Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)Polycyclic presentation:
2^8|a^2=f,c^2=d,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^c=eg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130798, #130801, #130810, … |
| maximal normal subgroup of | #130798, #130801, #130810, … |
| maximal quotient of | #135251, #135280, #135316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5854);