#506120

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

429 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,25,32)(19,27,35,46)(20,26,37,44)(21,38,34,48)(22,30,41,47)(23,29,39,43)(24,42,36,31)(28,33,45,40);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,33,24,34,41,37,25,35,39,40,36,21,22,20)(18,26,30,38,31,28,43,46,32,44,47,48,42,45,29,27);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,21,36,40,39,35,25,37,41,34,24,33,23,19)(18,27,29,45,42,48,47,44,32,46,43,28,31,38,30,26);(2,7)(6,8)(17,21,39,37,24,19,22,40,25,34,23,20,36,35,41,33)(18,28,47,27,31,44,29,38,32,45,30,46,42,26,43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,c^2=e,d^2=fi,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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413316);