Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1526 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
561 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8,9,10)(11,12,15,18,16,13,14,17);(3,4)(5,8)(6,10)(7,9)(11,13,16,12)(14,18,15,17);(3,5,7,6)(4,8,9,10)(11,14,16,15)(12,17,13,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | #106039, #107355, #108307, … |
| maximal normal subgroup of | #106039, #107355, #108307, … |
| maximal quotient of | C2×#7564, C2×#7570, #160437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10625);