#491579

order	512
nilpotency class	4
isoclinism	isoclinic class of #491572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

600 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398775);