#275962

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 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)(9,10,13,16,12,14,20,23)(11,17,19,22,18,24,21,15)(25,26,30,32,29,33,28,34)(27,35,38,40,37,31,36,39);(2,5)(6,7)(10,14)(15,22)(16,23)(17,24)(25,27,29,37)(26,31,33,35)(28,36,30,38)(32,39,34,40);(9,11,12,18)(10,15,14,22)(13,21,20,19)(16,24,23,17)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(10,14)(15,22)(16,23)(17,24)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183158);