Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(10,14)(11,15)(12,18)(16,24)(17,22)(19,23)(20,21);(1,2,3,4)(6,8)(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14);(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #93915, #93919, #152521, … |
| maximal normal subgroup of | #93915, #93919, #152521, … |
| maximal quotient of | #140722, #140766, #141201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6223);