#506095

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, 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

436 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=fh,d^2=eh,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, 413291);