Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×#388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C26 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(3,4);(1,2)(3,4);(6,8)(10,11)(13,14,16,15);(9,10,12,11)(13,15,16,14);(5,6)(7,8)(9,11,12,10)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#388 |
| maximal subgroup of | #97073, #97075, #105570, … |
| maximal normal subgroup of | #97073, #97075, #105570, … |
| maximal quotient of | C22×#1434, C22×#1436, C22×#1437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27004);