#290906

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

712 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)(9,10,14,17)(11,19,22,16)(12,13,18,15)(20,23,24,21)(28,36)(30,38)(32,37)(35,40);(1,3,4,8)(2,5,6,7)(9,11,14,22)(10,16,17,19)(12,20,18,24)(13,21,15,23)(25,26)(27,33)(28,37)(29,31)(30,35)(32,36)(34,39)(38,40);(9,12,15,10,14,18,13,17)(11,20,21,19,22,24,23,16)(25,27,29,34)(26,30)(28,35,36,40)(31,38)(32,33)(37,39);(2,6)(5,7)(9,13,14,15)(10,12,17,18)(11,21,22,23)(16,20,19,24)(27,34)(30,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=h,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=dhi,g^c=gh

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, 198102);