Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(1,2,4,6)(3,7,8,5)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132059, #132060, #132061, … |
| maximal normal subgroup of | #132059, #132060, #132061, … |
| maximal quotient of | #121983, #121984, #122021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4376);