Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#1890 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1051 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 247 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);(4,6)(7,8)(9,12)(10,17,18,13)(11,14)(15,19,21,22)(16,20);(3,4,5,6)(7,9,11,16)(8,12,14,20)(10,15,18,21)(13,19,17,22);(7,10)(8,13)(9,15)(11,18)(12,19)(14,17)(16,21)(20,22)64 conjugacy classes — view the full character table.
| maximal subgroup of | #146594, #146600, #146602, … |
| maximal normal subgroup of | #146594, #146600, #146602, … |
| maximal quotient of | C2×#8966, C2×#8968, #193875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13946);