#328470

order	512
nilpotency class	3
isoclinism	isoclinic class of #326959
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	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,18)(20,23)(22,24)(26,30)(29,36)(33,40)(35,38);(2,5)(6,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(3,7)(6,8)(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19)(25,27)(26,31)(28,34)(29,38)(30,39)(32,40)(33,37)(35,36);(1,3,4,7)(2,6,5,8)(27,34)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 235666);