#486728

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
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	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393924);