#294812

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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)(4,5)(6,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19)(25,26,30,34)(27,35,37,39)(28,33,29,32)(31,36,40,38);(2,5)(6,7)(10,14)(15,22)(16,21)(17,24)(27,36)(28,29)(31,39)(32,33)(35,40)(37,38);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)(25,27,28,38,30,37,29,36)(26,31,32,35,34,40,33,39);(1,3,4,8)(2,6,5,7)(10,14)(15,22)(16,21)(17,24)(27,37)(31,40)(35,39)(36,38)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202008);