Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2103 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #400 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,8,13,9,24)(5,14,16,26,18,29,17,10)(6,19,21,33,22,11,23,28)(15,27,30,34,31,32,20,25);(3,5,9,16,8,18,7,17)(4,10,24,14,13,26,12,29)(6,20,23,15,22,30,21,31)(11,27,33,34,19,32,28,25);(3,6,7,21,8,22,9,23)(4,11,12,28,13,19,24,33)(5,15,16,30,18,31,17,20)(10,25,14,27,26,34,29,32)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137217, #137218, #137219, … |
| maximal normal subgroup of | #137217, #137218, #137219, … |
| maximal quotient of | C2×#10687, C2×#10689, C2×#10688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16740);