#506107

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

438 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)(17,18,23,29)(19,31,35,26)(20,37,33,25)(21,28,38,42)(22,27,32,45)(24,41,40,30)(34,47,39,44)(36,46,48,43);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,34,24,36,38,20,23,35,32,39,40,48,21,33)(18,25,28,43,30,44,45,26,29,37,42,46,41,47,27,31);(11,15)(13,16)(17,20,38,48,40,34,22,35,23,33,21,36,24,39,32,19)(18,26,45,47,41,43,28,37,29,31,27,44,30,46,42,25);(1,3,4,8)(2,6,5,7)(17,19,32,39,24,36,21,33,23,35,22,34,40,48,38,20)(18,25,42,46,30,44,27,31,29,37,28,43,41,47,45,26)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=fh,d^2=ehi,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

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, 413303);