#489124

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

679 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)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,24,31)(19,32,36,45)(20,37,33,47)(21,27,42,30)(22,43,23,44)(25,35,46,34)(26,38,48,39)(28,41,29,40);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,24,36)(18,25,31,46)(20,21,33,42)(22,38,23,39)(26,30,48,27)(28,47,29,37)(32,44,45,43)(34,41,35,40);(17,20)(18,26)(19,21)(22,35)(23,34)(24,33)(25,30)(27,46)(28,45)(29,32)(31,48)(36,42)(37,44)(38,40)(39,41)(43,47);(17,21,24,42)(18,27,31,30)(19,33,36,20)(22,40,23,41)(25,26,46,48)(28,44,29,43)(32,47,45,37)(34,39,35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396320);