#506101

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

451 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:

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

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,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, 413297);