#252896

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(26,31)(27,35)(28,30)(32,36)(37,38)(39,40);(1,3,4,8)(2,5,6,7)(10,14)(15,23)(16,21)(17,24)(25,26,29,31)(27,36,37,40)(28,33,30,34)(32,35,39,38);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(25,27,29,37)(26,32,31,39)(28,38,30,35)(33,40,34,36);(2,6)(5,7)(10,14)(11,18)(16,21)(19,22)(25,28,29,30)(26,33,31,34)(27,35,37,38)(32,36,39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=h,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160092);