#294494

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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

841 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)(11,19)(12,21)(13,17)(14,23)(15,18)(16,20)(22,24)(25,26,29,32)(27,30,34,38)(28,31,36,39)(33,40,35,37);(2,5)(6,7)(9,11,13,20)(10,16,17,19)(12,15,22,14)(18,21,23,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,38)(33,36)(39,40);(9,12,14,20,13,22,15,11)(10,16,18,21,17,19,23,24)(25,28)(26,31)(27,33,34,35)(29,36)(30,37,38,40)(32,39);(1,3,4,8)(2,6,5,7)(10,17)(16,19)(18,23)(21,24)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=eh,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,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201690);