#506096

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

470 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:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=f,c^2=fh,d^2=eh,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, 413292);