#258362

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
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

825 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)(11,18)(12,20)(13,17)(14,15)(16,19)(21,24)(22,23)(25,26,28,33)(27,35,37,31)(29,32,30,34)(36,40,38,39);(2,5)(6,7)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24)(25,27,28,37)(26,31,33,35)(29,38,30,36)(32,40,34,39);(9,12,13,21)(10,16,17,23)(11,19,14,22)(15,20,18,24)(26,32)(27,36)(29,30)(33,34)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(10,17)(15,18)(16,23)(20,24)(26,33)(31,35)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gi,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, 165558);