Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1950 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,11,14,12,15,10,19)(9,16,17,13,18,21,20,22);(7,9,11,17,12,18,10,20)(8,13,14,21,15,22,19,16);(3,4,5,6)76 conjugacy classes — view the full character table.
| maximal subgroup of | #109655, #109656, #109849, … |
| maximal normal subgroup of | #109655, #109656, #109849, … |
| maximal quotient of | C2×#9601, C2×#9602, C2×#9603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14164);