#498748

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,27,23,28,36,43,21,24,35,37,20,26,34,42)(19,29,32,44,31,41,45,48,30,25,38,40,33,39,46,47);(1,2)(3,8)(4,5)(6,7)(11,16)(13,15)(18,24)(19,30)(26,28)(27,37)(31,33)(32,38)(42,43)(45,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,21,30)(18,25,24,29)(20,33,23,31)(22,32,35,38)(26,41,28,39)(27,40,37,44)(34,46,36,45)(42,48,43,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,23)(18,26,24,28)(19,31,30,33)(22,34,35,36)(25,39,29,41)(27,42,37,43)(32,45,38,46)(40,47,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405944);