#499244

order	512
nilpotency class	4
isoclinism	isoclinic class of #498821
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	#60685	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

820 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:

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

Polycyclic presentation:

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

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, 406440);