#496078

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403274);