#489966

order	512
nilpotency class	4
isoclinism	isoclinic class of #489701
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 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

740 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397162);