#386534

order	512
nilpotency class	3
isoclinism	isoclinic class of #219696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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)(4,7)(6,8)(9,10,15,21,16,22,14,20)(11,23,13,17,26,34,25,19)(12,27,30,38,31,37,32,18)(24,36,28,35,40,33,29,39);(1,3)(2,4)(5,8)(6,7)(9,11,16,26)(10,17,22,19)(12,28,31,29)(13,15,25,14)(18,35,38,39)(20,34,21,23)(24,32,40,30)(27,36,37,33);(1,4,6,5)(2,7,8,3)(9,12,15,32,16,31,14,30)(10,18,20,27,22,38,21,37)(11,24,13,29,26,40,25,28)(17,33,23,39,19,36,34,35);(1,5,6,4)(2,7,8,3)(9,13,14,11,16,25,15,26)(10,19,20,34,22,17,21,23)(12,29,30,24,31,28,32,40)(18,36,27,35,38,33,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=efg,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=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, 293730);