#6802577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#32904	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1827 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)(17,18,23,32,22,26,21,25)(19,20,27,29,24,31,28,30);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19,22,24)(18,20,26,31)(21,28,23,27)(25,30,32,29);(11,16)(13,15)(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,28)(27,32);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,25,26,32)(19,27,24,28)(20,29,31,30);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709773);