#372531

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
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

1192 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,25,29,35)(14,20)(15,21)(16,23)(17,28,36,40)(18,30)(19,32)(22,37,26,33)(24,31)(27,34,39,38);(1,3,5,4)(2,6,7,8)(9,11,14,23,15,24,32,18)(10,16,19,13,21,30,20,31)(12,26,40,38,29,22,28,34)(17,33,25,39,36,37,35,27);(1,4,5,3)(2,6,7,8)(9,12,14,28,15,29,32,40)(10,17,19,35,21,36,20,25)(11,22,23,38,24,26,18,34)(13,27,30,37,31,39,16,33);(2,7)(6,8)(9,13,14,30,15,31,32,16)(10,18,19,11,21,23,20,24)(12,27,28,37,29,39,40,33)(17,34,35,22,36,38,25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279727);