Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2856 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
461 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 461 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 — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,22);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,22)(19,24);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,23,18,21,24,20,22);(1,4,5,3)(2,7,8,6)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^8|a^2=eg,b^2=eh,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #101881, #255416, #255426, … |
| maximal normal subgroup of | #101881, #255416, #255426, … |
| maximal quotient of | #316154, #316155, #316156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20962);