#252008

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

812 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,17)(14,19)(15,20)(16,22)(18,23)(21,24)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,17)(13,20)(14,21)(16,23)(18,24)(19,22)(26,30)(29,36)(33,40)(35,38);(9,12)(10,16)(11,19)(13,15)(14,24)(17,21)(18,20)(22,23)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(2,6)(5,7)(10,17)(12,15)(18,19)(22,24)(26,30)(27,34)(32,37)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159204);