Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19573 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1636 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);(3,4,8,12)(5,13,15,17)(6,16,9,10)(7,18,14,11)(20,24)(21,28)(22,25)(27,32)(29,30)(31,34);(3,5,14,10,8,15,7,16)(4,9,18,17,12,6,11,13)(19,20,25,32,23,27,22,24)(21,26,28,34,29,33,30,31);(3,6)(4,10)(5,11)(7,17)(8,9)(12,16)(13,14)(15,18)(19,21,23,29)(20,26,27,33)(22,30,25,28)(24,31,32,34);(5,15)(6,9)(10,16)(13,17)(21,29)(26,33)(28,30)(31,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261870);