Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
213 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 213 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) = 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)(14,18)(15,21)(17,24)(19,27)(20,23)(22,28)(25,26);(6,9)(10,12)(13,14,16,18)(15,19,22,26)(17,20,23,24)(21,25,28,27);(5,7,8,11)(6,10,9,12)(13,15,16,22)(14,19,18,26)(17,25,23,27)(20,28,24,21)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #130815, #130816, #130817, … |
| maximal normal subgroup of | #130815, #130816, #130817, … |
| maximal quotient of | #135253, #135282, #135318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5856);