Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,23)(22,30)(24,29)(28,32)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125319, #125429, #142071, … |
| maximal normal subgroup of | #125319, #125429, #142071, … |
| maximal quotient of | #392496, #393127, #393136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26249);