Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) = #72321180 (order 4096) · Inn(G) = #357 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #357 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(4,9,10,25)(5,13)(6,18)(7,20)(11,29,28,33)(12,30,22,27)(14,19,26,24)(15,21)(16,34)(17,23)(31,32);(3,4)(5,14)(6,19)(7,22)(8,10)(9,23)(11,18)(12,20)(13,25)(15,28)(16,29)(17,26)(21,24)(27,32)(30,34)(31,33)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1418, #151972, #152161, … |
| maximal normal subgroup of | C4×#1418, #151972, #152161, … |
| maximal quotient of | C2×#7385, C2×#7386, #150063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6609);