#6820388

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=f,e^2=hi,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, 6727584);