#215239

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,g^c=ghi

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, 122435);