#506143

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

410 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,25,32)(19,27,34,46)(20,26,37,45)(21,38,36,43)(22,30,39,48)(23,29,40,42)(24,41,35,31)(28,47,44,33);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,33,24,21,39,37,25,34,40,47,35,36,22,20)(18,26,30,43,31,44,42,46,32,45,48,38,41,28,29,27);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,36,35,47,40,34,25,37,39,21,24,33,23,19)(18,27,29,28,41,38,48,45,32,46,42,44,31,43,30,26);(2,7)(6,8)(17,21,23,37,24,34,39,47,25,36,40,20,35,19,22,33)(18,28,30,27,31,26,42,43,32,44,48,46,41,45,29,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413339);