#489609

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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 ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

688 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(18,24)(20,33)(21,35)(22,37)(25,40)(26,38)(27,34)(28,44)(29,30)(31,47)(36,45)(39,42)(41,43)(46,48);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,18)(19,25)(20,34)(21,26)(22,38)(23,27)(24,35)(28,37)(29,40)(30,46)(31,48)(32,42)(33,44)(36,41)(39,45)(43,47);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,19,23,32)(18,25,27,42)(20,29,35,45)(21,36,33,30)(22,31,37,47)(24,39,34,40)(26,41,44,46)(28,43,38,48);(2,9)(3,12)(8,10)(13,14)(17,20,22,21,23,35,37,33)(18,26,28,34,27,44,38,24)(19,29,31,36,32,45,47,30)(25,41,43,40,42,46,48,39)

Polycyclic presentation:

2^9|c^2=i,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,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, 396805);