#8552373

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8459569);