#489156

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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 ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

655 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)(19,31)(20,33)(21,26)(22,41)(23,44)(24,29)(25,30)(27,40)(28,39)(32,46)(34,43)(35,47)(36,45)(37,42)(38,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,20)(21,34)(22,42)(23,45)(24,32)(25,35)(26,37)(27,46)(28,43)(29,36)(30,38)(31,41)(33,40)(39,48)(44,47);(17,20)(18,19)(21,36)(22,43)(23,46)(24,37)(25,38)(26,32)(27,45)(28,42)(29,34)(30,35)(31,39)(33,44)(40,47)(41,48);(17,21,25,24)(18,26,30,29)(19,32,35,34)(20,36,38,37)(22,39,23,40)(27,41,28,44)(31,46,47,43)(33,45,48,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fhi,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, 396352);