#293237

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 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)(11,17)(12,18)(14,20)(16,21)(19,23)(22,24)(25,26,29,33)(27,31,34,30)(28,36,38,35)(32,40,37,39);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,14,19,22)(12,16,18,21)(17,20,23,24)(25,27,29,34)(26,30,33,31)(28,37,38,32)(35,39,36,40);(9,11,18,23,13,19,12,17)(10,14,21,24,15,22,16,20)(25,28,31,40,29,38,30,39)(26,32,27,35,33,37,34,36);(2,6)(5,7)(10,15)(11,19)(16,21)(17,23)(26,33)(27,34)(32,37)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200433);