#6819052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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

1468 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)(17,18,23,26)(19,28,30,22)(20,31,29,24)(21,27,32,25);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,32)(21,29)(23,30)(24,25)(26,28)(27,31);(9,12,13,11)(10,14,15,16)(17,20,32,19,23,29,21,30)(18,24,25,22,26,31,27,28);(1,3,4,7)(2,6,5,8)(17,21,23,32)(18,25,26,27)(19,29,30,20)(22,24,28,31);(2,5)(6,8)(17,22,21,24,23,28,32,31)(18,19,27,20,26,30,25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6726248);