Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#2844 |
| 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 | |
935 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);(3,4,6,8)(5,7,9,10)(11,12,15,14)(13,16,17,18);(5,9)(7,10)(12,14)(16,18)(19,20,22,24)(21,25,26,23);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21,22,26)(20,23,24,25);(3,5)(4,7)(6,9)(8,10)(11,14,15,12)(13,16,17,18)52 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29676);