Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2822 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1023 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(8,9)(12,13)(15,16,20,22)(17,23,25,18)(19,21,29,30)(24,28,27,26);(4,6)(7,8,11,9)(10,12,14,13)(15,17,20,25)(16,18,22,23)(19,27,29,24)(21,28,30,26);(3,4)(5,6)(7,9,11,8)(10,12,14,13)(15,18)(16,17)(19,28)(20,23)(21,27)(22,25)(24,30)(26,29);(7,10,11,14)(8,12,9,13)(15,19,20,29)(16,21,22,30)(17,24,25,27)(18,26,23,28)52 conjugacy classes — view the full character table.
| maximal subgroup of | #111251, #111938, #131378, … |
| maximal normal subgroup of | #111251, #111938, #131378, … |
| maximal quotient of | C2×#14629, C2×#14631, C2×#14634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29671);