#349407

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256603);