Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8,12,16)(9,17,19,22)(10,15,11,14)(13,20,21,18);(3,4)(5,6)(7,9)(8,13)(10,20)(11,18)(12,19)(14,22)(15,17)(16,21);(4,6)(7,9)(8,13)(10,18)(11,20)(12,19)(14,17)(15,22)(16,21)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137136, #137138, #137206, … |
| maximal normal subgroup of | #137136, #137138, #137206, … |
| maximal quotient of | C2×#9964, C2×#9965, C2×#9967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16393);