#72285481

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623666	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #20514 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3583 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)(4,20)(5,26)(6,31)(7,12)(8,13)(9,15)(10,25)(11,29)(16,35)(17,49)(18,34)(19,43)(21,32)(22,40)(23,37)(24,38)(27,59)(28,60)(30,39)(33,51)(36,57)(41,44)(42,50)(45,58)(46,63)(47,52)(48,62)(53,56)(54,61)(55,64);(1,3,6,15,8,19,7,18)(2,9,12,14,13,34,31,43)(4,21,25,47,24,16,23,44)(5,17,30,45,29,50,27,51)(10,35,20,52,37,32,38,41)(11,33,39,42,26,58,59,49)(22,55,28,54,56,48,57,46)(36,64,53,63,60,62,40,61);(3,15)(4,22,24,56)(5,27)(6,7)(9,14)(10,36,37,60)(11,39)(12,31)(16,46,21,54)(18,19)(20,40,38,53)(23,28,25,57)(26,59)(29,30)(32,61,35,63)(33,58)(34,43)(41,62,52,64)(44,48,47,55)(45,51);(1,4,7,23,8,24,6,25)(2,10,31,38,13,37,12,20)(3,16,18,47,19,21,15,44)(5,28,27,22,29,57,30,56)(9,32,43,52,34,35,14,41)(11,40,59,60,26,53,39,36)(17,46,51,48,50,54,45,55)(33,63,49,64,58,61,42,62);(4,23,24,25)(5,29)(10,20,37,38)(11,26)(16,47,21,44)(17,50)(22,28,56,57)(27,30)(32,41,35,52)(33,58)(36,40,60,53)(39,59)(42,49)(45,51)(46,55,54,48)(61,62,63,64)

Polycyclic presentation:

2^10|b^2=h,c^2=f,d^2=hj,e^2=f,g^2=i,h^2=j,i^2=j,b^a=bh,c^b=chj,d^a=dh,d^b=dj,d^c=dg,e^d=ei,f^d=fj,g^a=gj,g^c=gi,g^d=gi,g^e=gj,h^a=hj,h^c=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286129
maximal normal subgroup of	#72286129
maximal quotient of	#72287387, #72309435, #72309511, …

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