#6986036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907945
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1100 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=fg,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei

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, 6893232);