#495441

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, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,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, 402637);