#252566

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

825 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)(9,10,13,17)(11,12,20,15)(14,19,23,18)(16,24,21,22)(25,26,31,27)(28,36,39,40)(29,33,30,34)(32,37,35,38);(1,3,4,8)(2,5,6,7)(9,11,13,20)(10,15,17,12)(14,16,23,21)(18,24,19,22)(25,27,31,26)(28,36,39,40)(29,34,30,33)(32,37,35,38);(9,12)(10,16)(11,18)(13,15)(14,24)(17,21)(19,20)(22,23)(25,28,31,39)(26,32,27,35)(29,37,30,38)(33,36,34,40);(2,6)(5,7)(10,17)(12,15)(18,19)(22,24)(25,29,31,30)(26,33,27,34)(28,37,39,38)(32,36,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,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,f^c=fh,g^c=gh

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, 159762);