#489973

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

764 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)(11,21)(12,17)(13,26)(14,31)(15,20)(16,22)(18,29)(19,28)(23,34)(24,32)(25,35)(27,37)(30,38)(33,40)(36,39);(1,2)(3,6)(4,8)(5,7)(9,11,15,12)(10,16,20,32)(13,27,30,40)(14,25,28,23)(17,26,21,38)(18,35,36,34)(19,37,31,33)(22,39,24,29);(1,2)(3,6)(4,8)(5,7)(9,12,15,11)(10,17,20,21)(13,24,30,22)(14,25,28,23)(16,26,32,38)(18,33,36,37)(19,34,31,35)(27,29,40,39);(1,3,5,4)(2,6,7,8)(11,12)(16,32)(17,21)(22,24)(23,25)(27,40)(33,37)(34,35)

Polycyclic presentation:

2^9|b^2=i,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, 397169);