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×#18651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,7,10)(5,9,6,8)(12,17)(13,21)(14,15)(19,20)(22,23)(24,25);(4,8)(5,6)(9,10)(11,12,16,20)(13,18,22,26)(14,17,15,19)(21,24,23,25);(11,13)(12,18)(14,21)(15,23)(16,22)(17,24)(19,25)(20,26);(3,5,7,6)(4,8,10,9)(11,14,16,15)(12,19,20,17)(13,21,22,23)(18,25,26,24)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255425);