Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
749 conjugacy classes of subgroups; only the 339 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 339 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(10,13)(14,16)(17,18)(19,21)(20,23)(22,24);(5,6,8,7)(18,19)(22,23);(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22);(1,3,4,2)(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=f,d^2=efg,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6781);