Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
595 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,21,23,24);(6,8)(10,11)(12,16)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(9,12)(10,14)(11,15)(13,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #125831, #126182, #126183, … |
| maximal normal subgroup of | #125831, #126182, #126183, … |
| maximal quotient of | #154715, #154723, #154978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8969);