Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 4 generators on 32 points:
(1,2)(3,11)(4,15,18,8)(5,9)(6,10)(7,14)(12,26,27,23)(13,21)(16,30,31,20)(17,29,32,24)(19,25)(22,28);(1,3,5,14)(2,7,9,11)(4,16,17,12)(6,13,19,28)(8,23,24,20)(10,22,25,21)(15,26,29,30)(18,31,32,27);(1,4,5,17)(2,8,9,24)(3,12,14,16)(6,18,19,32)(7,20,11,23)(10,15,25,29)(13,27,28,31)(21,26,22,30);(2,9)(3,13)(7,21)(8,24)(10,25)(11,22)(12,27)(14,28)(15,29)(16,31)(20,26)(23,30)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #213886, #213887, #213888, … |
| maximal normal subgroup of | #213886, #213887, #213888, … |
| maximal quotient of | #215608, #215611, #215612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14815);