#506146

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ehi,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,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, 413342);