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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,13)(14,17)(15,18)(16,20)(21,24)(22,23)(25,26,30,31)(27,32,35,28)(29,37,38,33)(34,36,40,39);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,21)(13,14)(15,20)(17,22)(18,19)(23,24)(26,31)(28,32)(33,37)(36,39);(11,20)(14,24)(16,19)(17,21)(25,27,30,35)(26,28,31,32)(29,34,38,40)(33,36,37,39);(1,3,4,7)(2,6,5,8)(9,12,15,22)(10,13,18,23)(11,21,20,17)(14,19,24,16)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,13)(10,12)(11,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(9,14)(10,17)(11,13)(12,19)(15,24)(16,22)(18,21)(20,23)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7908171);