#296755

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=eh,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, 203951);