Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10631 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
181 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(9,10,13,15,14,18,12,17)(11,19,21,24,20,23,22,16);(1,2)(3,5)(4,7)(6,8)(10,15)(11,20)(12,13)(16,23)(17,18)(19,24);(2,5)(3,8)(6,7)(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23)Polycyclic presentation:
2^8|a^2=e,c^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126295, #126349, #126387, … |
| maximal normal subgroup of | #126295, #126349, #126387, … |
| maximal quotient of | #105209, #105210, #105215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3619);