#506098

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

444 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:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=f,c^2=fhi,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, 413294);