#72320717

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#360748
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1965 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=eij,f^2=i,i^2=j,b^a=bf,c^a=cfh,c^b=cfij,d^a=dg,e^b=ej,e^c=ej,f^a=fi,f^c=fi,g^c=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320733, #72320921, #72343942
maximal normal subgroup of	#72320733, #72320921, #72343942

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