Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#2369 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1363 conjugacy classes of subgroups; only the 903 normal subgroups are shown.
This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 903 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,6,5)(7,8,11,10)(9,13,14,12)(16,19)(20,21);(3,5,6,4)(8,10)(12,13)(15,16)(17,21)(18,19)(20,22);(7,9,11,14)(8,12,10,13);(7,10,11,8)(9,13,14,12)(15,17,18,22)(16,20,19,21)100 conjugacy classes — view the full character table.
| maximal subgroup of | #97592, #106171, #108336, … |
| maximal normal subgroup of | #97592, #106171, #108336, … |
| maximal quotient of | C2×C4⋊C42, C2×#13818, C4×#2369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27131);