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 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,14)(13,20)(15,18)(17,21)(19,22)(23,24)(26,31)(29,39)(32,38)(33,35);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,26,30,31)(27,32,36,38)(28,35,37,33)(29,34,39,40);(10,12)(11,18)(17,20)(19,24)(26,31)(28,37)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,39,33);(1,4,5,3)(2,8,6,7)(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,28,30,37)(26,33,31,35)(27,34,36,40)(29,38,39,32);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,34)(27,35)(28,38)(30,39)(31,40)(32,37)(33,36)Polycyclic presentation:
2^9|b^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7623006);