Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2,3,6,4,7,8,5)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24);(2,5)(3,8)(6,7)(9,11,13,18)(10,14)(12,17,20,23)(15,21)(16,22)(19,24);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=g,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #131293, #131294, #131344, … |
| maximal normal subgroup of | #131293, #131294, #131344, … |
| maximal quotient of | #99828, #99832, #99846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2192);