Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | #3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
397 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(2,5)(3,7)(6,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(3,7)(5,8)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=f,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=dg,e^b=eg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93705, #137269, #137270, … |
| maximal normal subgroup of | #93705, #137269, #137270, … |
| maximal quotient of | #96944, #96947, #96949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1308);