#326723

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1050 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,22,12)(13,18,14,19)(17,23,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,36)(35,38)(39,40);(2,5)(6,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,24)(19,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,36)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,16)(10,17,20,21)(11,19,22,18)(13,23,14,24)(30,36)(37,40);(3,7)(6,8)(9,13,15,14)(10,18,20,19)(11,21,22,17)(12,23,16,24)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233919);