Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#2132 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| socle | C22 | C2×#99 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
653 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 653 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) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,7,12,8,13,5,10)(6,11,14,18,15,9,16,17);(4,9)(6,15)(10,17)(11,13)(12,18)(14,16);(3,5,8,7)(4,10,13,12)(6,14,15,16)(9,17,11,18)52 conjugacy classes — view the full character table.
| maximal subgroup of | #146619, #146620, #146624, … |
| maximal normal subgroup of | #146619, #146620, #146624, … |
| maximal quotient of | C2×#11431, C2×#11447, C2×#11475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26353);