#380901

order	512
nilpotency class	3
isoclinism	isoclinic class of #219518
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

896 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)(3,6)(4,5)(7,8)(9,10)(11,22,26,34)(12,27,28,37)(13,18)(14,19)(15,21)(16,25,35,40)(17,24,38,31)(20,32)(23,33)(29,39)(30,36);(2,5)(6,8)(9,11,14,25,15,26,32,40)(10,16,20,34,21,35,19,22)(12,13,24,29,28,30,31,23)(17,36,37,39,38,18,27,33);(1,3)(2,6)(4,7)(5,8)(9,12,15,28)(10,17,21,38)(11,23,26,29)(13,25,30,40)(14,31,32,24)(16,33,35,39)(18,22,36,34)(19,37,20,27);(3,7)(6,8)(9,13,14,29,15,30,32,23)(10,18,19,39,21,36,20,33)(11,24,25,28,26,31,40,12)(16,27,22,38,35,37,34,17)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 288097);