Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#2801 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1111 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 463 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(5,9)(11,12,14,16)(13,17,18,15)(20,21)(24,25);(4,7)(8,10)(19,20,23,21)(22,25,26,24);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,20)(22,25,26,24);(3,5,6,9)(4,8,7,10)(11,13,14,18)(12,15,16,17)(19,22,23,26)(20,24,21,25)64 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2801⋊C3 |
| maximal subgroup of | C4×#2801, #528973, #529047, … |
| maximal normal subgroup of | C4×#2801, #528973, #529047, … |
| maximal quotient of | C2×#14434, C2×#14459, C2×#14504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45247);