Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,6)(5,12,15,18)(7,11,8,16)(10,17,13,14)(21,27)(22,29)(24,30)(26,32)(28,33)(31,34);(3,5,8,14,9,15,7,17)(4,10,11,12,6,13,16,18)(19,20)(21,24)(22,26)(23,25)(27,30)(28,31)(29,32)(33,34);(3,6,9,4)(5,13,15,10)(7,11,8,16)(12,14,18,17)(19,21,22,27,23,28,29,33)(20,24,26,30,25,31,32,34);(3,7,9,8)(4,11,6,16)(5,14,15,17)(10,18,13,12)(20,25)(24,31)(26,32)(30,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264580);