#6836883

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=fi,e^2=gh,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, 6744079);