Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2644 |
| 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 487 normal subgroups are shown.
This group has 1111 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);(4,7)(8,9)(11,12)(13,16)(14,17)(15,18)(19,20,23,25)(21,26,22,24);(3,4,6,7)(5,8,10,9)(11,13)(12,14)(15,17)(16,18)(19,21,23,22)(20,24,25,26);(3,4)(5,9)(6,7)(8,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,26)(23,25);(3,5,6,10)(4,8,7,9)(11,14,15,16)(12,17,18,13)(19,22,23,21)(20,24,25,26)64 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2644⋊C3 |
| maximal subgroup of | #97039, #375634, #375644, … |
| maximal normal subgroup of | #97039, #375634, #375644, … |
| maximal quotient of | C2×#14260, C2×#14266, C2×#14331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27605);