#418335

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1719 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)(12,25)(13,14)(16,29)(17,31)(19,27)(21,36)(22,23)(26,28)(30,35)(32,33)(37,38)(39,40);(5,6)(7,8)(9,10)(11,20)(12,17)(13,27)(14,19)(15,18)(16,22)(21,35)(23,29)(24,34)(25,31)(26,33)(28,32)(30,36)(37,40)(38,39);(2,4)(9,11,15,24)(10,16,18,29)(12,21,26,37)(13,23,14,22)(17,30,33,39)(19,34,27,20)(25,38,28,36)(31,35,32,40);(5,7,8,6)(9,12,15,26)(10,17,18,33)(11,21,24,37)(13,25,14,28)(16,30,29,39)(19,32,27,31)(20,35,34,40)(22,36,23,38)

Polycyclic presentation:

2^9|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, 325531);