#490910

order	512
nilpotency class	4
isoclinism	isoclinic class of #490873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

485 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:

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

Polycyclic presentation:

2^9|b^2=egi,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398106);