Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #379 |
| prime radical | #3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
432 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 81 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)(3,7,8,5)(4,6)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,2)(3,5)(4,6)(7,8)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=f,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,e^b=eg,f^b=fg,d^c=dg52 conjugacy classes — view the full character table.
| Fitting subgroup of | #8317⋊C3 |
| maximal subgroup of | #137277, #137278, #137279, … |
| maximal normal subgroup of | #137277, #137278, #137279, … |
| maximal quotient of | #96944, #96945, #96946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1305);