#506144

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	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

427 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)(17,18,25,32)(19,27,38,47)(20,39,35,26)(21,41,37,45)(22,42,34,30)(23,43,33,29)(24,44,40,31)(28,48,46,36);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,33,48,24,37,34,20,25,38,23,36,40,21,22,35)(18,26,42,41,31,28,43,47,32,39,30,45,44,46,29,27);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,37,40,48,23,19,25,35,34,21,24,36,33,38)(18,27,29,46,44,45,30,39,32,47,43,28,31,41,42,26);(2,7)(6,8)(17,21,33,35,24,19,34,48,25,37,23,20,40,38,22,36)(18,28,42,47,31,39,43,45,32,46,30,27,44,26,29,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,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, 413340);