Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(1,2,3,4)(10,15)(11,19)(12,13)(16,23)(17,18)(22,24);(5,6,7,8)(9,10,13,17,14,18,12,15)(11,16,21,23,19,24,20,22);(9,11,12,20,14,19,13,21)(10,16,15,22,18,24,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129952, #129953, #138012, … |
| maximal normal subgroup of | #129952, #129953, #138012, … |
| maximal quotient of | #108548, #108549, #108552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3815);