Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60986 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,30)(23,28)(25,27)(26,31)(29,33)(32,34);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21,24,20)(22,26,32,29)(23,28,25,27)(30,33,34,31);(5,12)(8,11)(13,18)(15,17)(19,22)(20,26)(21,29)(23,31)(24,32)(25,33)(27,30)(28,34);(4,9)(8,11)(10,16)(15,17)(19,23,24,25)(20,27,21,28)(22,31,32,33)(26,30,29,34);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(19,23,24,25)(20,28,21,27)(22,31,32,33)(26,34,29,30)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483026);