#6631342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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:

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

Polycyclic presentation:

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

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, 6538538);