Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#2804 |
| 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 | |
1327 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1327 conjugacy classes of subgroups — too many to draw. Only its 503 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)(11,12,14,15)(13,16,17,18)(19,20)(21,24)(22,23)(25,26);(4,7)(8,10)(12,15)(13,17)(20,22)(21,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,25)(20,24,22,26);(3,5,6,9)(4,8,7,10)(11,13)(12,16)(14,17)(15,18)(19,22,23,20)(21,26,25,24)64 conjugacy classes — view the full character table.
| maximal subgroup of | #97543, #131459, #149479, … |
| maximal normal subgroup of | #97543, #131459, #149479, … |
| maximal quotient of | C2×#14440, C2×#14457, C2×#14471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45251);