#10086332

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9993528);