#491577

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
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

572 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)(12,17)(13,26)(14,29)(15,19)(16,23)(18,27)(21,34)(22,35)(24,37)(25,36)(28,30)(31,39)(32,38)(33,40);(1,2,3,5,4,6,7,8);(9,11,21,38,14,22,13,23,15,24,27,39,12,25,28,40)(10,16,30,35,17,32,18,20,19,33,26,36,29,31,34,37);(2,6)(5,8)(9,12,15,14)(10,17,19,29)(11,22,24,25)(13,21,28,27)(16,31,33,32)(18,26,34,30)(20,35,37,36)(23,39,40,38)

Polycyclic presentation:

2^9|b^2=e,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, 398773);