#72307134

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#17595
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁵
socle	C₂	C₂×#17595
prime radical	#10575319	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=hij,c^2=hij,d^2=f,f^2=i,g^2=ij,h^2=j,i^2=j,b^a=bj,d^a=dg,d^c=di,e^d=ehij,f^a=fh,f^c=fj,f^e=fj,g^a=gij,g^c=gj,g^d=ghi,g^f=gj,h^a=hj,h^d=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307146, #72307207, #72307482
maximal normal subgroup of	#72307146, #72307207, #72307482

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