Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,15)(13,18)(14,20)(16,21)(19,23)(22,24)(25,26)(27,30)(28,38)(29,40)(31,35)(32,37)(33,39)(34,36);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(25,27)(26,32)(28,29)(30,35)(31,37)(33,40)(34,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,12,13,21)(10,15,18,16)(11,19,20,24)(14,22,17,23)(25,28,31,39)(26,33,35,38)(27,29,37,36)(30,34,32,40);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,19)(13,21)(14,23)(15,18)(17,22)(20,24)(25,28)(26,33)(27,36)(29,37)(30,40)(31,39)(32,34)(35,38);(9,11)(10,17)(12,19)(13,20)(14,18)(15,23)(16,22)(21,24)(25,29,31,36)(26,34,35,40)(27,28,37,39)(30,33,32,38);(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,22)(23,24)(25,30,31,32)(26,27,35,37)(28,34,39,40)(29,38,36,33)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627620);