Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19643 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| 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 | |
1970 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)(5,11)(6,14)(7,9)(8,13)(10,15)(12,17)(16,18)(19,20)(21,28)(22,26)(23,32)(24,27)(25,30)(29,33)(31,34);(3,5,6,13,7,12,15,18)(4,8,10,11,9,16,14,17);(4,9)(8,16)(10,14)(11,17)(19,21,23,30,24,31,22,29)(20,25,26,28,27,33,32,34);(5,12)(8,16)(11,17)(13,18)(19,22,24,23)(20,26,27,32)(21,29,31,30)(25,28,33,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264323);