Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2634 |
| 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 | |
1143 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(8,11)(12,14)(15,16,20,18)(17,23,25,21)(19,22,27,26)(24,29,30,28);(4,6)(8,11)(12,14)(15,17,20,25)(16,21,18,23)(19,24,27,30)(22,28,26,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18,20,16)(17,21,25,23)(19,22,27,26)(24,29,30,28);(7,9,10,13)(8,12,11,14)(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)64 conjugacy classes — view the full character table.
| maximal subgroup of | #111624, #112907, #115190, … |
| maximal normal subgroup of | #111624, #112907, #115190, … |
| maximal quotient of | C2×#14270, C2×#14289, C2×#14298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27588);