Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62159 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6250 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);(3,4,7,10)(5,11,12,8)(6,14,15,9)(13,16,18,17)(19,20,23,29)(21,30,32,25)(22,27,24,28)(26,33,34,31);(3,5)(4,8)(6,13)(7,12)(9,16)(10,11)(14,17)(15,18)(19,21)(20,25)(22,33)(23,32)(24,31)(26,27)(28,34)(29,30);(19,21)(20,26)(22,31)(23,32)(24,33)(25,27)(28,30)(29,34);(5,12)(8,11)(13,18)(16,17)(19,22,23,24)(20,27,29,28)(21,31,32,33)(25,34,30,26);(3,6,7,15)(4,9,10,14)(5,13,12,18)(8,16,11,17)(19,22,23,24)(20,28,29,27)(21,31,32,33)(25,26,30,34)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486605);