Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3186 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
324 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,3,7,5,8,4,6)(9,10,13,17,14,18,12,15)(11,19,21,16,22,23,20,24);(2,6)(3,4)(7,8)(10,15,18,17)(16,19,24,23);(2,7)(3,4)(6,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9277, #198711, #198716, … |
| maximal normal subgroup of | C2×#9277, #198711, #198716, … |
| maximal quotient of | #99919, #99921, #99927, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2265);