Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
252 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 252 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8,5,3,4,7)(10,14)(11,17)(12,20)(16,22)(18,19)(21,23);(1,3,5,2)(4,7,6,8)(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152181, #152485, #153298, … |
| maximal normal subgroup of | #152181, #152485, #153298, … |
| maximal quotient of | #100113, #100119, #100867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2423);