#489984

order	512
nilpotency class	4
isoclinism	isoclinic class of #489727
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 ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 397180);