Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1868 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,11,8,12,18,24)(5,9,15,21,14,20,28,32)(6,10,17,23,16,22,29,33)(13,25,27,34,26,19,30,31);(3,5,7,15,8,14,18,28)(4,9,11,21,12,20,24,32)(6,13,17,27,16,26,29,30)(10,19,23,31,22,25,33,34);(3,6)(4,10)(5,13)(7,17)(8,16)(9,19)(11,23)(12,22)(14,26)(15,27)(18,29)(20,25)(21,31)(24,33)(28,30)(32,34)64 conjugacy classes — view the full character table.
| maximal subgroup of | #147333, #147334, #147335, … |
| maximal normal subgroup of | #147333, #147334, #147335, … |
| maximal quotient of | C2×#8851, #192027, C2×#8852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13879);