Finite Group Databasebeta

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#72318249

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139793
commutatorC23×D4C22×C4
Frattini, cosocleC2×#535C23
prime radicalC2×#3019C22
LCSG ▷ C23×D4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1522 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1522 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,15,19,41)(4,21)(5,27)(6,14)(7,13,8,12)(9,18,36,28)(10,26)(11,16,43,30)(17,37,48,62)(20,45,32,60)(22,53,56,64)(23,58)(24,40,25,39)(29,59)(31,33)(34,47,61,57)(35,51,44,50)(38,46,42,54)(49,52,55,63);(1,3,16,18,6,20,30,51)(2,9,33,45)(4,22,54,55,26,57,46,48)(5,28,7,19,31,50,8,32)(10,37,58,53)(11,41,13,44)(12,36,43,60)(14,35,27,15)(17,25,47,23,49,24,56,29)(21,52,59,34)(38,61,39,63)(40,62,42,64);(1,4)(2,10)(3,17,20,49)(5,29)(6,26)(7,24)(8,25)(9,34,35,53)(11,42)(12,40)(13,39)(14,21)(15,37,45,52)(16,46)(18,47,51,56)(19,48,32,55)(22,28,57,50)(23,31)(27,58)(30,54)(33,59)(36,61,44,64)(38,43)(41,62,60,63)

Polycyclic presentation:

2^10|a^2=f,b^2=dfg,c^2=gh,g^2=j,h^2=j,b^a=bd,c^a=cghj,c^b=cde,d^a=dg,d^b=dgi,d^c=dghj,e^a=eh,e^b=ehij,e^c=eghj,e^d=ej,f^b=fgj,f^d=fij,f^e=fij,g^a=gi,g^b=gij,g^d=gj,g^e=gj,g^f=gj,h^a=hi,h^b=hi,h^d=hj,h^e=hj,h^f=hj,i^a=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318267, #72318311, #72318389, …
maximal normal subgroup of#72318267, #72318311, #72318389, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.4*F.6*F.7);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);