Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19,22,24)(18,23);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24);(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=e,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129221, #129293, #129330, … |
| maximal normal subgroup of | #129221, #129293, #129330, … |
| maximal quotient of | #122225, #122231, #122237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4414);