Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1879 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 249 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:
(1,2,5,10,6,11,20,28,7,12,4,18,21,30,22,9)(3,13,15,26,16,29,23,27,17,25,14,31,24,32,19,8);(1,3)(2,8,12,27)(4,19)(5,23)(6,24)(7,17)(9,25,28,13)(10,29,18,32)(11,31,30,26)(14,22)(15,20)(16,21);(2,9)(4,20)(5,22)(6,21)(8,25)(10,30)(11,18)(12,28)(13,27)(14,23)(15,19)(16,24)(26,29)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13033, #494553, #495739, … |
| maximal normal subgroup of | C2×#13033, #494553, #495739, … |
| maximal quotient of | #136701, #136703, #136735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6021);