Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18602 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2914 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)(5,12)(6,10)(7,16)(8,11)(9,13)(14,18)(15,17)(20,24)(21,27)(22,29)(26,30)(28,31)(32,34);(3,5)(4,9)(6,13)(7,14)(8,15)(10,17)(11,18)(12,16)(19,20)(21,25)(22,30)(23,26)(24,29)(27,32)(28,33)(31,34);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,17,18,12)(19,21,23,28)(20,25,26,33)(22,31,29,27)(24,32,30,34);(3,6,8,7)(4,10,11,16)(5,14,15,13)(9,12,18,17)(20,26)(24,30)(25,33)(32,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255359);