Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 977 conjugacy classes of subgroups — too many to draw. Only its 479 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)(5,8)(6,7)(9,10)(11,12,15,18)(13,19,20,26)(14,23,21,17)(16,24,25,22);(4,7)(8,10)(13,20)(16,25)(19,26)(22,24);(11,13,21,24,15,20,14,22)(12,16,23,26,18,25,17,19);(3,5,6,9)(4,8,7,10)(13,20)(16,25)(19,26)(22,24)64 conjugacy classes — view the full character table.
| maximal subgroup of | #111392, #111395, #112640, … |
| maximal normal subgroup of | #111392, #111395, #112640, … |
| maximal quotient of | C2×#20824, C2×#21287, C2×#21409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53680);