Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1173 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1173 conjugacy classes of subgroups — too many to draw. Only its 487 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) = 64 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,26);(4,7)(8,10)(11,13,15,21)(12,16,18,24)(14,20,22,26)(17,23,19,25);(3,5,6,9)(4,8,7,10)(12,17)(14,22)(16,23)(18,19)(20,26)(24,25);(12,18)(16,24)(17,19)(23,25)64 conjugacy classes — view the full character table.
| maximal subgroup of | #114871, #114974, #198134, … |
| maximal normal subgroup of | #114871, #114974, #198134, … |
| maximal quotient of | C2×#20701, C2×#20845, C2×#20910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53496);