#506141

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,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, 413337);