Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19001 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,13)(6,12)(7,11)(8,10)(9,14)(15,18)(16,17)(19,20)(21,27)(22,30)(23,25)(24,28)(26,32)(29,34)(31,33);(3,5)(4,9)(6,16)(7,14)(8,15)(10,17)(11,18)(12,13)(19,21,23,29)(20,24,25,33)(22,31,32,28)(26,34,30,27);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18)(22,32)(26,30)(27,34)(28,31);(4,10)(9,17)(11,12)(13,18)(20,25)(24,33)(26,30)(27,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258579);