#489151

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
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×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

676 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)(19,31,35,48)(20,29,23,26)(21,38,22,41)(25,42,45,33)(27,37,28,36)(32,46,34,47)(39,43,40,44);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,25)(20,32)(21,39)(22,40)(23,34)(24,35)(26,43)(27,46)(28,47)(29,44)(30,45)(31,38)(33,36)(37,42)(41,48);(17,19)(18,25)(20,32)(21,40)(22,39)(23,34)(24,35)(26,43)(27,47)(28,46)(29,44)(30,45)(31,41)(33,37)(36,42)(38,48);(17,20,24,23)(18,26,30,29)(19,32,35,34)(21,36,22,37)(25,43,45,44)(27,38,28,41)(31,47,48,46)(33,40,42,39)

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,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, 396347);