Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C25 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
633 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 633 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93101, #93119, #93137, … |
| maximal normal subgroup of | #93101, #93119, #93137, … |
| maximal quotient of | #97066, #97072, #97074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1531);