Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(2,4)(5,6,7,8)(10,14)(11,18)(12,15)(16,24)(17,23)(20,21);(1,2,3,4)(9,10,15,23,13,17,12,14)(11,19,18,16,21,22,20,24);(9,11,12,18,13,21,15,20)(10,16,14,22,17,24,23,19)Polycyclic presentation:
2^8|a^2=g,b^2=dh,c^2=e,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93961, #93962, #136147, … |
| maximal normal subgroup of | #93961, #93962, #136147, … |
| maximal quotient of | #121349, #121352, #121366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4319);