#292141

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

852 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(26,31)(27,35)(28,30)(32,36)(37,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,26)(27,36)(28,33)(29,31)(30,34)(32,38)(35,39)(37,40);(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,27,28,35,29,37,30,38)(26,32,34,40,31,39,33,36);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,29,30)(26,33,31,34)(27,35,37,38)(32,36,39,40)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,g^c=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, 199337);