#500105

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	#60459	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

983 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)(11,21)(12,17)(13,16)(14,19)(15,23)(18,28)(20,27)(22,30)(24,38)(25,34)(26,32)(29,39)(31,37)(33,40)(35,36);(2,5)(6,8)(9,11,12,23)(10,15,17,21)(13,24,26,37)(14,22,27,25)(16,31,32,38)(18,29,33,35)(19,34,20,30)(28,36,40,39);(1,3,4,7)(2,6,5,8)(10,16)(11,22)(13,28)(14,27)(15,29)(17,32)(18,20)(19,33)(21,35)(23,25)(24,37)(26,40)(30,38)(31,34);(2,5)(6,8)(10,17)(15,21)(16,32)(18,33)(19,20)(29,35)(30,34)(31,38)

Polycyclic presentation:

2^9|b^2=i,c^2=ei,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, 407301);