#293473

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

827 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)(4,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23)(25,26,28,31)(27,30,35,39)(29,37,36,32)(33,38,34,40);(2,5)(6,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,17)(18,21)(23,24)(25,27,28,35)(26,30,31,39)(29,33,36,34)(32,40,37,38);(9,12,13,21)(10,15,16,23)(11,18)(14,22)(17,24)(19,20)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(1,3,4,8)(2,6,5,7)(10,16)(12,21)(14,17)(18,20)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=ehi,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,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, 200669);