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 | C2×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C26 |
| commutator, Frattini, cosocle | C2 | C27 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
29593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(3,4)(5,10)(6,11)(7,19)(8,9)(12,20)(13,23)(14,28)(15,21)(16,25)(17,31)(18,22)(24,29)(26,32)(27,34)(30,33);(4,9)(10,21)(11,22)(12,19)(23,25)(24,28)(26,31)(33,34);(3,5,8,15)(4,10,9,21)(6,16,18,13)(7,14,20,29)(11,25,22,23)(12,24,19,28)(17,30,32,27)(26,34,31,33);(3,6,8,18)(4,11,9,22)(5,13,15,16)(7,17,20,32)(10,23,21,25)(12,26,19,31)(14,27,29,30)(24,33,28,34);(3,4)(5,10)(6,11)(7,12)(8,9)(13,23)(14,24)(15,21)(16,25)(17,26)(18,22)(19,20)(27,33)(28,29)(30,34)(31,32);(3,7)(4,12)(5,14)(6,17)(8,20)(9,19)(10,24)(11,26)(13,27)(15,29)(16,30)(18,32)(21,28)(22,31)(23,33)(25,34)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56090);