#506119

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

420 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)(19,27,37,48)(20,38,35,26)(21,40,42,47)(22,43,34,30)(23,44,33,29)(24,45,39,31)(28,36,46,41);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,33,41,24,21,34,20,25,37,23,36,39,42,22,35)(18,26,43,47,31,46,44,48,32,38,30,40,45,28,29,27);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,21,39,41,23,19,25,35,34,42,24,36,33,37)(18,27,29,28,45,40,30,38,32,48,44,46,31,47,43,26);(2,7)(6,8)(17,21,23,35,24,37,22,41,25,42,33,20,39,19,34,36)(18,28,30,48,31,26,29,40,32,46,43,27,45,38,44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413315);