Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
219 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 77 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)(3,5)(4,7)(6,8)(9,10,14,17,15,12,13,18)(11,19,21,16,23,24,22,20);(2,5)(3,8)(6,7)(9,11,14,21,15,23,13,22)(10,16,17,24,12,20,18,19);(9,12)(10,15)(11,20)(13,17)(14,18)(16,23)(19,21)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8902, #192902, #192903, … |
| maximal normal subgroup of | C2×#8902, #192902, #192903, … |
| maximal quotient of | #99403, #99406, #99409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1890);