Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61723 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6344 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,24,28)(21,29,30,34)(22,27,23,26)(25,32,33,31);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(21,30)(25,33)(29,34)(31,32);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,21,24,30)(20,25,28,33)(22,31,23,32)(26,29,27,34);(5,13)(8,11)(12,18)(15,17)(19,22,24,23)(20,26,28,27)(21,31,30,32)(25,29,33,34);(4,10)(8,11)(9,16)(15,17)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485089);