Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C2×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C26 |
| commutator, Frattini, cosocle | C2 | C27 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
30285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 7 generators on 18 points:
(1,2);(3,4)(5,9)(6,13)(7,8)(10,14)(11,15)(12,18)(16,17);(4,8)(9,15)(10,13)(16,18);(5,11)(9,15)(12,17)(16,18);(3,5)(4,9)(6,12)(7,11)(8,15)(10,16)(13,18)(14,17);(3,4)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14)(17,18);(3,6)(4,10)(5,12)(7,14)(8,13)(9,16)(11,17)(15,18)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56089);