#6834564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42988	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=i,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,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6741760);