Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C23 | C22×C8 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,7)(9,10,12,11)(13,14,15,17,16,18,19,20);(1,2)(3,4)(5,6)(7,8)(9,11,12,10);(2,4)(5,7,8,6)(10,11)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,g^2=h,b^a=bd,c^a=ce,c^b=cf88 conjugacy classes — view the full character table.
| Fitting subgroup of | #11298⋊C3 |
| maximal subgroup of | C2×#11298, #358322, #358323, … |
| maximal normal subgroup of | C2×#11298, #358322, #358323, … |
| maximal quotient of | #121067, #121076, #121077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4286);