Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 711 conjugacy classes of subgroups — too many to draw. Only its 199 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)(3,4)(5,6)(7,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(2,4)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19);(6,8)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93518, #93519, #106058, … |
| maximal normal subgroup of | #93518, #93519, #106058, … |
| maximal quotient of | #105497, #105500, #105501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3711);