Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2064 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
503 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8,11,18,13,17,12,16)(9,15,19,14,10,21,20,22);(3,4)(5,6)(7,9,11,19,13,10,12,20)(8,14,16,15,17,22,18,21);(4,6)(7,10,13,9)(8,15,17,21)(11,19,12,20)(14,18,22,16)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2064, #359397, #377550, … |
| maximal normal subgroup of | C4×#2064, #359397, #377550, … |
| maximal quotient of | C2×#10613, C2×#10614, #242364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16588);