Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,13,14,15)(17,18,19,21,20,22,23,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg64 conjugacy classes — view the full character table.
| maximal subgroup of | #97570, #97571, #97572, … |
| maximal normal subgroup of | #97570, #97571, #97572, … |
| maximal quotient of | #96238, #96732, #96911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 561);