#490858

order	512
nilpotency class	4
isoclinism	isoclinic class of #490801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

431 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dgi,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398054);