Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #380 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
231 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,3,5,4,6,7,8)(9,10,14,17,15,12,13,23)(11,18,21,19,22,24,20,16);(9,11,13,20,15,22,14,21)(10,16,17,18,12,19,23,24);(9,12)(10,15)(11,19)(13,23)(14,17)(16,22)(18,21)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^c=eg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #129522, #129523, #129565, … |
| maximal normal subgroup of | #129522, #129523, #129565, … |
| maximal quotient of | #98836, #98837, #98840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1592);