#489160

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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

657 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)(18,25)(19,31)(20,23)(21,22)(26,41)(27,36)(28,29)(30,40)(32,33)(34,48)(35,37)(38,47)(39,44)(42,45);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,24,30)(19,26,34,45)(20,29,23,27)(21,36,37,28)(22,25,35,40)(31,43,48,46)(32,41,38,42)(33,39,47,44);(17,19)(18,26)(20,32)(21,31)(22,33)(23,38)(24,34)(25,39)(27,42)(28,46)(29,41)(30,45)(35,47)(36,43)(37,48)(40,44);(17,20,24,23)(18,27,30,29)(19,32,34,38)(21,35,37,22)(25,28,40,36)(26,42,45,41)(31,47,48,33)(39,46,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396356);