Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C22 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
333 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 179 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:
(1,2,4,7,5,8,3,6)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,e^2=h,f^2=g,b^a=bd,d^a=dg,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#344, #105387, #105388, … |
| maximal normal subgroup of | C8×#344, #105387, #105388, … |
| maximal quotient of | #94916, #94917, #94932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 906);