Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2001 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1091 conjugacy classes of subgroups — too many to draw. Only its 239 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);(3,4)(5,6)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(7,8)(9,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(4,6)(8,10)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7078, #122561, C2×#7086, … |
| maximal normal subgroup of | C2×#7078, #122561, C2×#7086, … |
| maximal quotient of | C2×#10258, C2×#10259, #232923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16373);