Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2)(3,7,8,6)(4,5)(10,14)(11,17)(12,20)(16,22)(18,19)(21,23);(2,5)(6,7)(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=dgh,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130254, #130255, #130732, … |
| maximal normal subgroup of | #130254, #130255, #130732, … |
| maximal quotient of | #98840, #98841, #98846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1617);