Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1618 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
600 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12,16,19)(13,20,22,25)(14,18,15,23)(17,24,26,21);(3,4)(5,10)(6,8)(7,9)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(5,9)(7,10)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1618, #208883, #208907, … |
| maximal normal subgroup of | C4×#1618, #208883, #208907, … |
| maximal quotient of | C2×#7790, C2×#7793, C2×#7796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11108);