#328926

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

662 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ei,d^2=eh,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, 236122);