Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1991 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 287 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 — 4 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,23)(18,27)(19,21)(20,25)(22,24)(26,29)(28,30);(4,6)(8,11)(12,13)(15,17,24,30,19,26,18,25)(16,20,27,29,21,28,22,23);(7,9,10,14)(8,12,11,13)(16,21)(20,28)(22,27)(23,29)88 conjugacy classes — view the full character table.
| maximal subgroup of | #107761, #110429, #110431, … |
| maximal normal subgroup of | #107761, #110429, #110431, … |
| maximal quotient of | C2×#9871, C2×#9875, #206795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14305);