Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1653 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
473 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12,16,20)(13,21,24,25)(14,17,15,19)(18,22,26,23);(3,4,6,8)(5,10,9,7)(12,17)(14,15)(18,25)(19,20)(21,26)(22,23);(11,13,16,24)(12,18,20,26)(14,23,15,22)(17,25,19,21)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1653, #209723, #209771, … |
| maximal normal subgroup of | C4×#1653, #209723, #209771, … |
| maximal quotient of | C2×#7798, C2×#7768, C2×#7813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11488);