#492289

order	512
nilpotency class	4
isoclinism	isoclinic class of #492236
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

440 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)(3,9,10,16)(4,13,14,7)(6,11,15,12)(17,18,21,24)(19,28,30,45)(20,26,23,37)(22,34,33,47)(25,41,39,31)(27,32,40,36)(29,46,38,44)(35,42,48,43);(1,3)(2,6)(4,11)(5,10)(7,16)(8,15)(9,13)(12,14)(18,24)(25,39)(26,37)(27,40)(28,45)(34,47)(42,43)(44,46);(3,10)(6,15)(9,16)(11,12)(17,19)(18,25)(20,29)(21,30)(22,35)(23,38)(24,39)(26,42)(27,44)(28,34)(31,36)(32,41)(33,48)(37,43)(40,46)(45,47);(3,11)(4,14)(6,16)(7,13)(9,15)(10,12)(17,20,21,23)(18,26,24,37)(19,29,30,38)(22,32,33,36)(25,42,39,43)(27,34,40,47)(28,46,45,44)(31,35,41,48)

Polycyclic presentation:

2^9|a^2=gi,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,c^b=cg,d^b=dei,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399485);