Finite Group Databasebeta

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#72318380

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139773
commutatorC2×#127C22×C4
Frattini, cosocleC2×#535C23
prime radicalC2×#3021C22
LCSG ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1445 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1445 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,9)(4,20)(5,25)(6,12)(7,13)(8,14)(10,24)(11,28)(15,42)(16,46)(17,34)(18,33)(19,32)(21,52)(22,37)(23,41)(26,57)(27,39)(29,40)(30,45)(31,47)(35,49)(36,56)(38,48)(43,59)(44,50)(51,54)(53,61)(55,60)(58,64)(62,63);(1,3)(2,9)(4,15,22,49)(5,26,27,47)(6,18,7,17)(8,19)(10,35,41,42)(11,38)(12,34,13,33)(14,32)(16,29)(20,50,37,30)(21,53,56,62)(23,44,24,45)(25,31,39,57)(28,48)(36,58,52,59)(40,46)(43,55,64,54)(51,61,60,63);(1,4,5,21,8,24,27,55)(2,10,11,36)(3,15,16,43)(6,22,28,54,7,23,29,56)(9,30,31,58,32,42,57,61)(12,37,39,60)(13,41,25,52)(14,20,40,51)(17,49,47,64)(18,44,26,53)(19,45,48,62)(33,50,46,63,34,35,38,59)

Polycyclic presentation:

2^10|b^2=fgh,c^2=d,d^2=ij,g^2=j,h^2=j,b^a=bf,c^a=cfhij,c^b=ce,d^a=dg,d^b=dh,e^a=eg,e^b=egi,e^c=ehj,e^d=ei,f^c=fgi,f^d=fij,f^e=fi,g^a=gj,g^b=gi,g^c=gi,g^e=gj,g^f=gj,h^a=hj,h^b=hi,h^c=hij,h^e=hj,h^f=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318389, #72318472, #72319043, …
maximal normal subgroup of#72318389, #72318472, #72319043, …

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, 2, F.6*F.7*F.8);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
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.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*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> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);