Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1786 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 85 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)(3,13,16,28)(4,18)(5,10)(6,12)(7,14,24,31)(8,15,25,17)(9,20)(11,22)(19,29,32,23)(21,26)(27,30);(1,3)(2,7)(4,19)(5,15)(6,16)(8,9)(10,23)(11,29)(12,24)(13,26)(14,30)(17,22)(18,28)(20,31)(21,32)(25,27);(1,3)(2,8)(4,14)(5,17)(6,16)(7,9)(10,28)(11,13)(12,25)(15,22)(18,23)(19,30)(20,32)(21,31)(24,27)(26,29)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fh,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12750, #493556, #495739, … |
| maximal normal subgroup of | C2×#12750, #493556, #495739, … |
| maximal quotient of | #129522, #129565, #129789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5738);