Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = #72339316 (order 4096) · Inn(G) = #535 · Out(G) = D42 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2487 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,19)(3,12,20,23,13,26,15,8)(4,16,21,24,17,30,18,9)(14,28,31,32,27,22,29,25);(1,3,5,15,6,13,7,20)(2,8,10,12,11,23,19,26)(4,14,18,29,17,27,21,31)(9,22,16,25,24,28,30,32);(3,13)(4,17)(9,24)(15,20)(16,30)(18,21)(22,28)(25,32);(1,4,5,18,6,17,7,21)(2,9,10,16,11,24,19,30)(3,14,15,29,13,27,20,31)(8,22,12,25,23,28,26,32)Polycyclic presentation:
2^8|a^2=eg,b^2=e,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #486694, #486695, #500158, … |
| maximal normal subgroup of | #486694, #486695, #500158, … |
| maximal quotient of | #314967, #314968, #314969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20680);