#328346

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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)(9,10,13,18)(11,12,20,16)(14,23,15,19)(17,22,24,21)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(2,5)(6,8)(9,11)(10,16)(12,18)(13,20)(14,17)(15,24)(19,22)(21,23)(26,30)(29,36)(33,40)(35,38);(3,7)(6,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,21)(18,24)(20,23)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(1,3,4,7)(2,6,5,8)(10,18)(12,16)(19,23)(21,22)(26,30)(27,34)(32,37)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235542);