Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#1719 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
425 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 425 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) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,8)(9,10)(11,12)(13,19)(14,16)(15,23)(17,21)(18,22)(20,25)(24,26);(3,4,8,10,6,9,5,7)(12,16)(17,24)(18,23)(19,25);(4,9)(7,10)(11,13,15,21,14,20,22,26)(12,17,18,19,16,24,23,25)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1719, #213118, #213125, … |
| maximal normal subgroup of | C4×#1719, #213118, #213125, … |
| maximal quotient of | C2×#8821, C2×#9826, C2×#9829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12599);