#6558474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(3,8)(4,6)(5,7)(17,18,23,27)(19,28,29,32)(20,25,24,31)(21,26,22,30);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(19,29)(20,24)(25,31)(28,32);(10,13)(14,16)(17,19,22,25,23,29,21,31)(18,24,26,28,27,20,30,32);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,25)(19,30)(21,28)(22,32)(23,24)(26,29)(27,31);(11,15)(14,16)(17,21,23,22)(18,26,27,30)(19,31,29,25)(20,32,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465670);