#291452

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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

737 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)(11,14)(12,20)(13,19)(15,17)(16,23)(18,21)(22,24)(25,26,31,34)(27,32,37,40)(28,29,33,30)(35,39,38,36);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,20,22,24)(13,19,15,17)(16,23,18,21)(25,27,31,37)(26,32,34,40)(28,38,33,35)(29,36,30,39);(9,12,13,23,14,22,15,21)(10,16,17,20,11,18,19,24)(25,28,30,26,31,33,29,34)(27,35,36,32,37,38,39,40);(2,6)(5,7)(9,13,14,15)(10,17,11,19)(12,21,22,23)(16,24,18,20)(25,29,31,30)(26,33,34,28)(27,36,37,39)(32,35,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,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=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198648);