Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C23×#23900 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C22×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
209 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,5)(3,7)(6,8)(9,10,13,16,14,17,12,21)(11,18,19,15,20,23,22,24);(1,2)(3,8)(4,6)(5,7)(9,11,13,19,14,20,12,22)(10,15,16,23,17,24,21,18);(3,7)(5,8)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136096, #136097, #136098, … |
| maximal normal subgroup of | #136096, #136097, #136098, … |
| maximal quotient of | #99313, #99314, #99318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1852);