#380900

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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)(3,6)(4,5)(7,8)(9,10)(11,20,24,35)(12,17)(13,18)(14,29)(15,19)(16,22,31,23)(21,32,37,40)(25,38,36,30)(26,33)(27,39)(28,34);(2,5)(6,8)(9,11,14,23,15,24,13,22)(10,16,18,20,19,31,29,35)(12,21,27,36,28,37,26,25)(17,30,33,32,34,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,25)(13,23)(14,22)(15,24)(17,32)(18,35)(19,31)(20,29)(21,26)(27,37)(28,36)(30,33)(34,40)(38,39);(3,7)(6,8)(9,12,13,26,15,28,14,27)(10,17,18,33,19,34,29,39)(11,21,22,25,24,37,23,36)(16,30,20,32,31,38,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 288096);