Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 183 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);(3,4,7,11,8,12,23,30)(5,13,16,31,14,27,21,33)(6,17,19,29,20,10,22,28)(9,24,26,18,25,32,34,15);(3,5)(4,9)(6,18)(7,21)(8,14)(10,27)(11,26)(12,25)(13,17)(15,20)(16,23)(19,24)(22,32)(28,33)(29,31)(30,34);(5,14)(9,25)(13,27)(15,18)(16,21)(24,32)(26,34)(31,33)46 conjugacy classes — view the full character table.
| maximal subgroup of | #146659, #146676, #146703, … |
| maximal normal subgroup of | #146659, #146676, #146703, … |
| maximal quotient of | C2×#11432, C2×#11448, C2×#11480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26354);