Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#2643 |
| 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 | |
1273 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1273 conjugacy classes of subgroups — too many to draw. Only its 487 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);(3,4,7,8)(5,6,9,10)(12,15)(16,17)(19,20)(21,24)(22,25)(23,26);(4,8)(5,9)(11,12,14,15)(13,16,18,17)(19,21)(20,23)(22,26)(24,25);(3,5)(4,6)(7,9)(8,10)(11,12)(13,17)(14,15)(16,18)(19,21)(20,24)(22,26)(23,25);(3,6)(4,9)(5,8)(7,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,23)(22,25)(24,26)64 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2643⋊C3 |
| maximal subgroup of | #375649, #381278, #391942, … |
| maximal normal subgroup of | #375649, #381278, #391942, … |
| maximal quotient of | C2×#14259, C2×#14265, C2×#14331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27553);