#489381

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 ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,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, 396577);