Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #389 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
155 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 155 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:
(1,2,4,7)(3,5,8,6)(9,10,13,18,14,19,23,17)(11,15,12,21,20,16,22,24);(2,5)(3,8)(6,7)(9,11,13,12,14,20,23,22)(10,15,18,21,19,16,17,24);(9,12,13,20,14,22,23,11)(10,16,17,21,19,15,18,24)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=e,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ch,d^b=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | #136360, #136372, #136373, … |
| maximal normal subgroup of | #136360, #136372, #136373, … |
| maximal quotient of | #101934, #101937, #101940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2619);