#418425

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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1685 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)(9,10,13,18)(11,21,24,16)(12,26,28,38)(14,20,15,19)(17,31,36,29)(22,33,40,35)(23,32,27,39)(25,34,30,37);(5,6)(7,8)(9,11)(10,16)(12,27)(13,24)(14,30)(15,25)(17,35)(18,21)(19,37)(20,34)(22,29)(23,28)(26,32)(31,40)(33,36)(38,39);(2,4)(9,12)(10,17)(11,22)(13,28)(14,31)(15,29)(16,32)(18,36)(19,38)(20,26)(21,39)(23,25)(24,40)(27,30)(33,34)(35,37);(5,7,8,6)(9,12)(10,17)(11,23)(13,28)(14,29)(15,31)(16,33)(18,36)(19,26)(20,38)(21,35)(22,25)(24,27)(30,40)(32,34)(37,39)

Polycyclic presentation:

2^9|a^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 325621);