#496082

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

466 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)(17,18,22,27,23,24,30,31,21,26,28,32,19,20,25,29);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,23)(18,24,26,20)(22,25,28,30)(27,31,32,29);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,21,23)(18,24,26,20)(22,25,28,30)(27,31,32,29);(1,4,5,3)(2,8,6,7)(17,20,28,31,23,18,25,32,21,24,22,29,19,26,30,27)

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^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403278);