#6843904

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

1467 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)(17,18,22,25)(19,20,29,24)(21,26,31,32)(23,27,30,28);(1,2)(3,7)(4,8)(5,6)(17,19,22,29)(18,24,25,20)(21,23,31,30)(26,28,32,27);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,30,32,22,24,23,26)(18,19,28,31,25,29,27,21);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,25)(20,24)(26,32)(27,28);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,21,30,19,22,31,23,29)(18,26,28,20,25,32,27,24)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=gi,e^2=fghi,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, 6751100);