#326879

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

924 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)(11,19)(12,21)(13,17)(14,20)(15,18)(16,22)(23,24)(25,26,29,31)(27,33,35,40)(28,36,37,32)(30,38,39,34);(2,5)(6,8)(9,11)(10,12)(13,21)(14,16)(15,24)(17,19)(18,22)(20,23)(28,37)(30,39)(32,36)(34,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,19)(10,16,17,24)(11,20,21,18)(14,23,15,22)(25,27,29,35)(26,30,31,39)(28,38,37,34)(32,40,36,33);(3,7)(6,8)(10,17)(12,19)(18,20)(22,23)(26,31)(30,39)(32,36)(33,40)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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, 234075);