Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) = #72313305 (order 2048) · Inn(G) = #386 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1869 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,6,12,7,10,24,27,8,11,25,29,5,22,14,31)(3,13,17,23,18,28,32,4,16,20,30,19,15,9,26,21);(1,3,14,26,5,15,25,30,8,16,24,32,7,18,6,17)(2,9,27,23,22,20,12,21,11,28,31,19,10,13,29,4);(1,4)(2,3)(5,23)(6,13)(7,19)(8,21)(9,14)(10,18)(11,16)(12,30)(15,22)(17,29)(20,25)(24,28)(26,27)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=eh,d^c=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129595, #129596, #129613, … |
| maximal normal subgroup of | #129595, #129596, #129613, … |
| maximal quotient of | #136580, #136588, #136596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5976);