#72307367

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346522
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=egj,d^2=hj,e^2=j,f^2=i,g^2=j,i^2=j,b^a=bhj,c^a=ce,c^b=cf,d^a=dhj,d^c=degh,e^b=eh,e^c=eh,e^d=ehj,f^a=fh,f^b=fi,f^c=fhij,f^d=fhj,f^e=fj,g^a=ghj,g^c=gh,g^d=ghj,g^e=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307431, #72307865, #72308079
maximal normal subgroup of	#72307431, #72307865, #72308079

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