#382717

order	512
nilpotency class	3
isoclinism	isoclinic class of #213788
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,21,31)(12,16,24,34)(13,26,28,17)(19,29,23,33)(20,36,27,30)(22,37,39,32)(25,40,38,35);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,22)(13,27)(14,21)(16,32)(17,36)(18,31)(19,25)(20,28)(23,38)(24,39)(26,30)(29,35)(33,40)(34,37);(2,6)(5,8)(9,12)(10,16)(11,19,21,23)(13,25)(14,24)(15,29,31,33)(17,35)(18,34)(20,22,27,39)(26,40)(28,38)(30,32,36,37);(9,11)(10,15)(12,23)(13,20)(14,21)(16,33)(17,30)(18,31)(19,24)(22,38)(25,39)(26,36)(27,28)(29,34)(32,40)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,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, 289913);