Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1864 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,5,11,7,12,6,24)(3,13,16,28,18,25,17,32)(4,19,21,29,23,9,22,10)(8,20,26,14,27,31,30,15);(1,3)(2,8,12,27)(4,20)(5,17)(6,16)(7,18)(9,25,19,13)(10,28,29,32)(11,30,24,26)(14,22)(15,21)(23,31);(2,9)(4,21)(5,6)(8,25)(10,24)(11,29)(12,19)(13,27)(14,31)(15,20)(16,17)(22,23)(26,28)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=egh,f^b=fh,g^b=gh,d^c=dgh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12953, #494332, #495739, … |
| maximal normal subgroup of | C2×#12953, #494332, #495739, … |
| maximal quotient of | #135906, #135930, #136223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5941);