#6826814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6734010);