#500106

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	#60458	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1006 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)(11,14)(12,23)(13,20)(15,31)(16,18)(17,27)(19,33)(21,37)(22,29)(24,32)(25,39)(26,38)(28,36)(30,35)(34,40);(2,5)(6,8)(9,11,14,10)(12,24,26,34)(13,20,16,18)(15,22,30,19)(17,28,21,25)(23,40,38,32)(27,39,37,36)(29,31,33,35);(1,3,4,7)(2,6,5,8)(9,12,14,26)(10,17,11,21)(13,25,16,28)(15,32,30,40)(18,34,20,24)(19,36,22,39)(23,35,38,31)(27,29,37,33);(2,5)(6,8)(9,13,14,16)(10,18,11,20)(12,25,26,28)(15,29,30,33)(17,34,21,24)(19,35,22,31)(23,36,38,39)(27,32,37,40)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,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, 407302);