#500578

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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	C₂²×C₈	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

801 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407774);