Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19368 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19381 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1628 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)(5,13,15,16)(6,10,9,18)(7,14,17,11)(19,20)(21,29)(22,28)(23,25)(24,27)(26,30)(31,34)(32,33);(3,5,7,10,8,15,17,18)(4,9,11,13,12,6,14,16)(20,25)(26,33)(27,28)(29,34);(3,6)(4,10)(5,14)(7,16)(8,9)(11,15)(12,18)(13,17)(19,21)(20,26)(22,30)(23,31)(24,32)(25,33)(27,29)(28,34);(5,15)(6,9)(10,18)(13,16)(19,22,23,24)(20,27,25,28)(21,30,31,32)(26,29,33,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261265);