#326861

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 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)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24)(25,26,31,35)(27,32,28,37)(29,34,30,39)(33,36,40,38);(2,5)(6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,19)(18,24)(25,27)(26,32)(28,31)(29,36)(30,38)(33,39)(34,40)(35,37);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(17,18)(21,22)(23,24)(25,28,31,27)(26,33,35,40)(29,38,30,36)(32,39,37,34);(3,7)(6,8)(9,12,13,14)(10,16,17,18)(11,21,22,20)(15,23,19,24)(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,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, 234057);