#495661

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

511 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,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, 402857);