#6828718

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,28,32)(20,29,31,25)(21,26,30,27);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,24,23,32)(20,30,31,21)(25,26,29,27);(2,5)(6,8)(10,14)(11,16)(17,20,23,29,22,31,18,25)(19,27,24,30,28,26,32,21);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,18,22,23)(19,24,28,32)(20,25,31,29)(21,27,30,26);(17,21,23,27,22,30,18,26)(19,25,24,20,28,29,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=fg,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6735914);