#6635626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632036
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	#24210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1602 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,24,22,26,20,25)(19,27,28,32,29,31,30,23);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,13,16);(3,7)(5,8)(9,12,14,13)(10,15,11,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(3,7)(5,8)(9,13,14,12)(10,15,11,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(2,6)(5,8)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,d^b=dfh,g^c=gi

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, 6542822);