#292361

order	512
nilpotency class	3
isoclinism	isoclinic class of #270875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

855 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)(11,17,19,24)(13,21,18,16)(15,23,22,20)(25,26,28,31)(27,30,35,39)(29,37,36,32)(33,38,34,40);(1,3,4,8)(2,5,6,7)(10,14)(15,22)(16,21)(17,24)(25,27)(26,30)(28,35)(29,33)(31,39)(32,40)(34,36)(37,38);(9,11,18,23,12,19,13,20)(10,15,21,24,14,22,16,17)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(2,6)(5,7)(10,14)(15,22)(16,21)(17,24)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fhi,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

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, 199557);