#486326

order	512
nilpotency class	4
isoclinism	isoclinic class of #486320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393522);