Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C25×#120 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
232 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(5,8)(9,10,14,18,15,19,13,21)(11,20,17,16,23,24,22,12);(1,2)(3,5)(4,6)(7,8)(9,11,14,17,15,23,13,22)(10,16,18,24,19,12,21,20);(1,3,4,7)(2,5,6,8)(9,12)(10,17)(11,21)(13,24)(14,20)(15,16)(18,23)(19,22)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eh,d^c=dh,e^c=eg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8875, #192456, #192458, … |
| maximal normal subgroup of | C2×#8875, #192456, #192458, … |
| maximal quotient of | #99406, #99407, #99408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1863);