#506084

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, 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

398 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,33)(20,38)(21,41)(22,30)(23,29)(24,31)(25,44)(26,34)(27,36)(28,40)(32,42)(35,48)(37,46)(39,47)(43,45);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,35,25,37,39,20,24,36,43,40,42,21,22,34)(18,26,30,41,32,28,45,27,31,38,47,46,44,48,29,33);(11,15)(13,16)(17,20,39,21,42,35,23,36,24,34,22,37,25,40,43,19)(18,27,45,48,44,41,30,38,31,33,29,28,32,46,47,26);(1,3,4,8)(2,6,5,7)(17,21,43,20,25,19,22,40,24,37,23,34,42,36,39,35)(18,28,47,33,32,38,29,41,31,48,30,27,44,26,45,46)

Polycyclic presentation:

2^9|a^2=gi,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^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, 413280);