Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #390 |
| prime radical | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,17)(12,14)(13,21)(15,20)(16,18)(19,23)(22,24);(2,6,8,7)(10,14)(15,22)(16,21)(17,23);(9,11,18,24,12,19,13,20)(10,15,21,23,14,22,16,17)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=eh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #92863, #138720, #138724, … |
| maximal normal subgroup of | #92863, #138720, #138724, … |
| maximal quotient of | #102101, #102103, #102105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2833);