Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#532 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator, Frattini, cosocle | C22 | C26 |
| socle | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
6439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(6,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(5,6,7,8);(9,11,12,15)(10,14,13,16)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#532 |
| maximal subgroup of | #184765, #184803, #184819, … |
| maximal normal subgroup of | #184765, #184803, #184819, … |
| maximal quotient of | C22×#2285, C22×#2293, C22×#2297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55654);