Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C23×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C2 | C27 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
29903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 22 points:
(3,4);(5,6);(1,2)(3,4)(5,6);(8,12)(10,18)(13,15)(16,22);(7,8,11,12)(9,15,17,13)(10,14,18,19)(16,21,22,20);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21);(7,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,21)(17,22)136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56087);