#418339

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1671 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)(6,7)(9,10,14,12)(11,21,24,17)(13,27,29,36)(15,20,16,19)(18,32,26,30)(22,31,35,25)(23,37,28,40)(33,39,34,38);(5,6)(7,8)(9,11)(10,17)(12,21)(13,28)(14,24)(15,31)(16,25)(18,34)(19,35)(20,22)(23,29)(26,33)(27,37)(30,38)(32,39)(36,40);(2,4)(9,12,14,10)(11,22,24,35)(13,26,29,18)(15,19,16,20)(17,31,21,25)(23,37,28,40)(27,32,36,30)(33,39,34,38);(5,7,8,6)(9,13)(10,18)(11,23)(12,26)(14,29)(15,30)(16,32)(17,33)(19,27)(20,36)(21,34)(22,37)(24,28)(25,38)(31,39)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 325535);