Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 18874368 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#2811 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1175 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);(3,4,7,5)(6,9,10,8)(11,12)(13,17)(14,15)(16,18)(20,21)(24,25);(4,5)(6,10)(12,15)(13,18)(19,20,23,21)(22,25,26,24);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,21,23,20)(22,25,26,24);(3,6,7,10)(4,8,5,9)(11,12)(13,16)(14,15)(17,18)(19,22)(20,24)(21,25)(23,26)64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45274);