Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3259 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1005 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 1005 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(5,9)(11,12)(13,16)(14,17)(15,19)(18,22)(20,24)(21,23)(25,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,20)(12,16,19,24)(14,21,22,26)(17,23,18,25);(4,7)(8,10)(12,17)(14,22)(16,23)(18,19)(21,26)(24,25);(3,5,6,9)(4,8,7,10)(13,20)(16,24)(21,26)(23,25)52 conjugacy classes — view the full character table.
| maximal subgroup of | #131362, #131372, #131396, … |
| maximal normal subgroup of | #131362, #131372, #131396, … |
| maximal quotient of | C2×#21897, C2×#21968, C2×#22126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53898);