Finite Group Databasebeta

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#72285718

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151462
commutator, Frattini, cosocle#1781C23
prime radicalC23×D4C2×D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

418 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 37 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=gij,d^2=j,e^2=g,b^a=be,c^a=cdfhij,c^b=cefgij,d^b=df,d^c=dh,e^a=efg,e^b=eghj,e^c=eghj,e^d=ehij,f^a=fh,f^c=fhij,f^e=fij,g^a=gij,g^b=gj,g^c=gj,g^d=gj,h^b=hi,h^c=hj,h^e=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285757, #72286122
maximal normal subgroup of#72285757, #72286122

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