#489587

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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

713 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)(17,18)(19,31)(20,26)(21,38)(22,40)(23,43)(24,30)(25,33)(27,42)(28,39)(29,41)(32,48)(34,47)(35,45)(36,46)(37,44);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19,24,20)(18,25,30,46)(21,36,39,33)(22,32,42,35)(23,37,41,34)(26,40,31,27)(28,44,38,47)(29,48,43,45);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20,24,19)(18,26,30,31)(21,32,39,35)(22,36,42,33)(23,37,41,34)(25,40,46,27)(28,45,38,48)(29,47,43,44);(2,9)(3,12)(8,10)(13,14)(17,21,23,22,24,39,41,42)(18,27,29,28,30,40,43,38)(19,32,34,36,20,35,37,33)(25,44,45,26,46,47,48,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396783);