#290926

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

830 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)(9,10,15,18)(11,16,21,23)(12,14,17,13)(19,22,20,24)(25,26,30,33)(27,35,38,32)(28,29,34,31)(36,37,39,40);(1,3,4,8)(2,5,6,7)(9,11,15,21)(10,16,18,23)(12,22,17,24)(13,19,14,20)(25,27)(26,32)(28,39)(29,37)(30,38)(31,40)(33,35)(34,36);(9,12,14,18,15,17,13,10)(11,16,20,22,21,23,19,24)(25,28,31,26,30,34,29,33)(27,36,37,32,38,39,40,35);(2,6)(5,7)(9,13,15,14)(10,12,18,17)(11,19,21,20)(16,22,23,24)(25,29,30,31)(26,28,33,34)(27,37,38,40)(32,39,35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,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, 198122);