#489129

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

682 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)(17,18,24,30)(19,31,35,46)(20,36,32,48)(21,39,42,27)(22,43,23,44)(25,34,47,33)(26,37,45,38)(28,41,29,40);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,24,35)(18,25,30,47)(20,21,32,42)(22,37,23,38)(26,27,45,39)(28,48,29,36)(31,44,46,43)(33,41,34,40);(17,20)(18,26)(19,21)(22,34)(23,33)(24,32)(25,27)(28,46)(29,31)(30,45)(35,42)(36,44)(37,40)(38,41)(39,47)(43,48);(17,21,24,42)(18,27,30,39)(19,32,35,20)(22,40,23,41)(25,45,47,26)(28,43,29,44)(31,36,46,48)(33,38,34,37)

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,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396325);