Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(3,4)(5,9)(6,13)(7,8)(10,14)(11,15)(12,18)(16,17)(19,20)(21,29)(22,27)(23,28)(24,25)(26,32)(30,34)(31,33);(5,11)(9,15)(12,17)(16,18)(20,25)(26,29)(27,28)(33,34);(4,8)(9,15)(10,13)(16,18)(19,20)(21,26)(22,27)(23,28)(24,25)(29,32)(30,33)(31,34);(3,4)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14)(17,18)(19,21)(20,26)(22,30)(23,31)(24,32)(25,29)(27,33)(28,34);(3,5)(4,9)(6,12)(7,11)(8,15)(10,16)(13,18)(14,17)(19,22,24,23)(20,27,25,28)(21,30,32,31)(26,33,29,34);(3,6)(4,10)(5,12)(7,14)(8,13)(9,16)(11,17)(15,18)(19,23,24,22)(20,28,25,27)(21,31,32,30)(26,34,29,33)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493397);