Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3511 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3056 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)(5,13,16,17)(6,10,7,11)(9,15,18,14)(19,20)(21,29)(22,25)(23,27)(24,28)(26,30)(31,34)(32,33);(3,5,8,16)(4,9,12,18)(6,14,7,15)(10,17,11,13)(20,25)(22,23)(26,33)(27,28)(29,34)(30,31);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,17,18,13)(19,21,24,32)(20,26,28,34)(22,30,23,31)(25,33,27,29);(3,6,8,7)(4,11,12,10)(5,15,16,14)(9,17,18,13)(19,22,24,23)(20,27,28,25)(21,30,32,31)(26,29,34,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255431);