#489140

order	512
nilpotency class	4
isoclinism	isoclinic class of #488895
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

684 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)(19,32,37,47)(20,30,23,27)(21,25,38,29)(22,41,39,28)(26,40,46,35)(33,44,36,42)(34,43,48,45);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,25)(20,23)(21,39)(22,38)(26,42)(27,41)(28,30)(29,31)(32,43)(33,36)(34,35)(40,48)(44,46)(45,47);(17,19)(18,26)(20,33)(21,40)(22,34)(23,36)(24,37)(25,42)(27,43)(28,47)(29,44)(30,45)(31,46)(32,41)(35,38)(39,48);(17,20,24,23)(18,27,31,30)(19,33,37,36)(21,22,38,39)(25,28,29,41)(26,43,46,45)(32,42,47,44)(34,35,48,40)

Polycyclic presentation:

2^9|a^2=ghi,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,d^b=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396336);