#506138

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

423 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,26)(21,39)(22,30)(23,29)(24,31)(25,42)(28,41)(32,37)(33,44)(34,46)(35,47)(36,48)(38,45)(40,43);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,33,41,25,36,22,20,24,35,23,34,37,21,40,38)(18,26,43,48,32,28,29,27,31,45,30,39,42,46,44,47);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,21,37,41,33,35,24,38,40,36,25,34,23,19)(18,27,29,46,42,48,43,45,31,47,44,28,32,39,30,26);(2,7)(6,8)(17,21,33,38,25,19,22,41,24,36,23,20,37,35,40,34)(18,28,43,27,32,45,29,39,31,46,30,47,42,26,44,48)

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

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, 413334);