Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63027 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32360 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);(3,4,9,11)(5,12,7,10)(6,15,17,8)(13,14,16,18)(19,20,24,27)(21,28,29,25)(22,23,26,32)(30,34,33,31);(3,5,9,7)(4,10,11,12)(6,13,17,16)(8,14,15,18)(21,29)(25,28)(30,33)(31,34);(3,5,9,7)(4,10,11,12)(6,13,17,16)(8,14,15,18)(19,21)(20,25)(22,30)(23,31)(24,29)(26,33)(27,28)(32,34);(3,6)(4,8)(5,13)(7,16)(9,17)(10,14)(11,15)(12,18);(3,7,9,5)(4,10,11,12)(6,16,17,13)(8,14,15,18)(19,22)(20,23)(21,30)(24,26)(25,31)(27,32)(28,34)(29,33);(3,8)(4,6)(5,14)(7,18)(9,15)(10,13)(11,17)(12,16)(19,23)(20,26)(21,31)(22,27)(24,32)(25,33)(28,30)(29,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493543);