#6561038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6468234);