Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#120 | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 324 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:
(1,2,5,7,4,3,6,8)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24);(1,3,5,8,4,2,6,7)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,24)(20,23);(2,3)(7,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #131160, #131162, #131164, … |
| maximal normal subgroup of | #131160, #131162, #131164, … |
| maximal quotient of | #99835, #99836, #99837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2097);