#6131096

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=fghi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6038292);