Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #389 |
| prime radical | C23 | C4×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
153 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,5,3,4,6,8,7)(10,15)(11,18)(12,13)(16,23)(17,21)(19,20);(1,3,5,6,4,7,8,2)(9,10)(11,16)(12,21)(13,15)(14,17)(18,22)(19,24)(20,23);(9,11,12,19,14,20,13,18)(10,16,21,24,17,23,15,22)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eg,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #135885, #135886, #135888, … |
| maximal normal subgroup of | #135885, #135886, #135888, … |
| maximal quotient of | #101929, #101930, #101931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2596);