#464191

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1785 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)(3,4)(5,6)(7,8)(9,10)(11,21)(12,18)(13,29)(14,31)(15,20)(16,24)(17,28)(19,30)(22,38)(23,32)(25,37)(26,36)(27,35)(33,39)(34,40);(9,11)(10,16,20,34)(12,26)(13,23)(14,24)(15,25)(17,32,18,33)(19,21,31,37)(22,28)(27,39)(29,36,35,38)(30,40);(2,4)(9,12,15,28)(10,17,20,18)(11,22,25,26)(13,30,27,14)(16,32,34,33)(19,29,31,35)(21,36,37,38)(23,24,39,40);(6,8)(9,13,14,12,15,27,30,28)(10,18,19,29,20,17,31,35)(11,23,24,26,25,39,40,22)(16,33,21,36,34,32,37,38)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 371387);