Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
809 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 809 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,14,18)(11,19,12,22)(13,24,20,17)(15,23,16,21);(5,6)(7,8)(9,11,13,21,14,12,20,23)(10,15,17,19,18,16,24,22);(1,2,3,4)(6,8)(11,12)(15,16)(19,22)(21,23);(9,12,13,23,14,11,20,21)(10,16,17,22,18,15,24,19)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #301895, #301897, #301917, … |
| maximal normal subgroup of | #301895, #301897, #301917, … |
| maximal quotient of | #298661, #298667, #298673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18772);