Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| automorphism | Aut(G) = #72314018 (order 4096) · Inn(G) = #1893 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,6,13,8,12,7,28)(3,14,17,20,16,32,18,24)(4,19,21,31,22,10,9,29)(5,23,25,30,27,15,26,11);(1,3,5,10,6,17,25,29,8,16,27,19,7,18,26,31)(2,9,30,32,28,22,23,20,12,21,11,14,13,4,15,24);(1,4)(2,10)(3,15)(5,24)(6,9)(7,21)(8,22)(11,18)(12,19)(13,31)(14,26)(16,23)(17,30)(20,27)(25,32)(28,29)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130240, #130241, #130288, … |
| maximal normal subgroup of | #130240, #130241, #130288, … |
| maximal quotient of | #137619, #137620, #137659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6064);