#6836308

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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)(11,16)(12,14)(13,15)(19,28)(21,31)(23,32)(25,29);(1,2)(3,6)(4,5)(7,8)(17,18,22,26)(19,29,28,25)(20,24,30,27)(21,23,31,32);(9,11,13,12)(10,14,15,16)(17,19,20,21,22,28,30,31)(18,23,27,29,26,32,24,25);(1,3,4,7)(2,6,5,8)(17,20,22,30)(18,24,26,27)(19,21,28,31)(23,25,32,29);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,21,20,28,22,31,30,19)(18,25,27,23,26,29,24,32)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=fi,e^2=ghi,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^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, 6743504);