Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | #2617 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 63 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)(4,7)(6,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19,22,24)(18,23);(2,5)(3,8)(6,7)(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,d^2=g,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efg,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #129741, #129742, #129757, … |
| maximal normal subgroup of | #129741, #129742, #129757, … |
| maximal quotient of | #122544, #122547, #122550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4680);