Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
| direct product | C2×C4×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6);(7,8,9,10)(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(11,13,14,17)(12,16,15,18)160 conjugacy classes — view the full character table.
| maximal subgroup of | #105484, #105495, #105496, … |
| maximal normal subgroup of | #105484, #105495, #105496, … |
| maximal quotient of | C2×C4×#376, C2×C42×C22⋊C4, C2×C42×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26978);