#6908826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
prime radical	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,28,25,24)(20,31,32,30)(21,26,23,29);(1,3,4,8)(2,5,6,7)(9,11,12,16)(10,13,14,15)(17,19,22,25)(18,24,27,28)(20,29,32,26)(21,31,23,30);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,23,28,22,32,21,24)(18,25,26,31,27,19,29,30);(2,6)(5,7)(10,14)(13,15)(17,19)(18,24)(20,26)(21,31)(22,25)(23,30)(27,28)(29,32);(3,8)(5,7)(17,21,22,23)(18,26,27,29)(19,30,25,31)(20,24,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=i,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^d=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, 6816022);