Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
571 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 571 conjugacy classes of subgroups — too many to draw. Only its 155 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 — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,23)(22,24);(9,11,12,19,13,18,21,24)(10,14,16,17,15,22,20,23);(1,2,3,4)(10,15)(14,22)(16,20)(17,23);(6,8)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^8|b^2=e,c^2=fh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125162, #125333, #138147, … |
| maximal normal subgroup of | #125162, #125333, #138147, … |
| maximal quotient of | #351682, #351683, #351690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26165);