#6819090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(11,14)(12,16)(13,15)(17,18,23,28)(19,22,27,24)(20,25,30,32)(21,29,31,26);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,27)(18,24,28,22)(20,21,30,31)(25,26,32,29);(9,12,13,11)(10,14,15,16)(17,20,21,27,23,30,31,19)(18,25,29,24,28,32,26,22);(1,3,4,7)(2,6,5,8)(17,21,23,31)(18,26,28,29)(19,30,27,20)(22,25,24,32);(2,5)(6,8)(17,22,31,32,23,24,21,25)(18,27,26,20,28,19,29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,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,d^c=di,e^c=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, 6726286);