#6826639

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
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(19,20)(21,29)(22,31)(23,28)(24,26)(25,30)(27,32);(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,25,28,20)(21,22,24,32)(26,31,29,27);(2,5)(6,8)(11,16)(13,15)(17,20,21,27,23,25,24,31)(18,19,29,32,28,30,26,22);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,26,28,29)(19,22,30,32)(20,27,25,31);(17,22,21,19,23,32,24,30)(18,27,29,25,28,31,26,20)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6733835);