#72307137

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8376
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8376
prime radical	#10576601	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3751 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:

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

Polycyclic presentation:

2^10|a^2=hij,d^2=e,e^2=i,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bj,c^a=cj,c^b=cj,d^a=df,d^b=dfgj,e^a=ei,e^b=eh,f^a=fi,f^b=fij,g^a=gj,g^b=gj,g^d=ghi,g^e=gj,h^a=hj,h^b=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307146, #72307431
maximal normal subgroup of	#72307146, #72307431

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