#6841708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,24,20)(21,25,29,32)(27,30,31,28);(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,26)(21,28)(22,24)(25,27)(29,30)(31,32);(2,6)(7,8)(10,14)(11,16)(17,20,26,19,22,23,18,24)(21,28,32,27,29,30,25,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(21,29)(25,32)(27,31)(28,30);(1,4,5,3)(2,8,6,7)(17,21,18,25,22,29,26,32)(19,27,20,28,24,31,23,30)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6748904);