#419177

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1861 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)(9,10,15,20,13,18,14,19)(11,21,26,33,24,16,25,22)(12,27,31,35,29,36,30,17)(23,37,39,32,28,40,38,34);(5,7,8,6)(9,11,13,24)(10,16,18,21)(12,28,29,23)(14,25,15,26)(17,34,35,32)(19,33,20,22)(27,37,36,40)(30,39,31,38);(2,4)(6,7)(9,10,13,18)(11,22,24,33)(12,17,29,35)(14,20,15,19)(16,25,21,26)(23,37,28,40)(27,31,36,30)(32,39,34,38);(6,7)(9,12,13,29)(10,17,18,35)(11,23,24,28)(14,31,15,30)(16,32,21,34)(19,27,20,36)(22,37,33,40)(25,39,26,38)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=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, 326373);