#489603

order	512
nilpotency class	4
isoclinism	isoclinic class of #489394
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396799);