Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #392 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
283 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(2,6)(3,7)(4,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,24)(22,23);(1,2,5,3)(4,7,8,6)(9,11,14,20)(10,15,17,23)(12,19,13,22)(16,24,21,18);(1,3,5,2)(4,6,8,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,18,23,24)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94646, #94647, #125887, … |
| maximal normal subgroup of | #94646, #94647, #125887, … |
| maximal quotient of | #102545, #102550, #102556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3002);