#6843722

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=g,e^2=fgh,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, 6750918);