Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2034 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 32 points:
(1,2)(3,11)(4,15)(5,20)(6,9)(7,13)(8,18)(10,17)(12,27)(14,24)(16,23)(19,31)(21,28)(22,29)(25,32)(26,30);(7,22)(11,27)(23,26)(24,28);(1,3,5,21,6,12,17,14)(2,7,10,26,9,22,20,23)(4,16,18,29,19,30,32,13)(8,24,15,27,25,28,31,11)Polycyclic presentation:
2^8|c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^c=dh,e^c=egh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13395, #499047, #499261, … |
| maximal normal subgroup of | C2×#13395, #499047, #499261, … |
| maximal quotient of | #144714, #144715, #144750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6383);