#486306

order	512
nilpotency class	4
isoclinism	isoclinic class of #486306
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 ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

949 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393502);