#489971

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
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

759 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)(11,22)(12,20)(13,16)(14,18)(15,21)(17,23)(19,28)(24,37)(25,36)(26,35)(27,32)(29,34)(30,33)(31,40)(38,39);(1,2)(3,6)(4,8)(5,7)(10,16)(12,14)(13,28)(17,31)(18,32)(19,20)(21,33)(22,37)(23,26)(24,39)(27,30)(29,40)(34,36)(35,38);(1,2)(3,6)(4,8)(5,7)(9,11,15,25)(10,17,21,35)(12,26,14,23)(13,29,30,24)(16,31,33,38)(18,36,20,22)(19,37,32,34)(27,39,28,40);(1,3,5,4)(2,6,7,8)(9,12,15,14)(10,18,21,20)(11,23,25,26)(13,27,30,28)(16,19,33,32)(17,22,35,36)(24,39,29,40)(31,34,38,37)

Polycyclic presentation:

2^9|c^2=i,d^2=h,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,d^c=di,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, 397167);