Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,15,17,13,18,14,19)(11,12,22,24,21,16,20,23);(1,3,5,8)(2,6,7,4)(9,11,13,21)(10,16,18,12)(14,20,15,22)(17,23,19,24);(10,17)(11,20)(14,15)(18,19)(21,22)(23,24);(1,4,5,6)(2,3,7,8)(9,12,13,16)(10,11,18,21)(14,24,15,23)(17,20,19,22)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125733, #125735, #125947, … |
| maximal normal subgroup of | #125733, #125735, #125947, … |
| maximal quotient of | #345520, #345523, #345528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24242);