Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
538 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,22)(13,18)(14,20)(15,19)(17,23)(21,24);(2,5)(6,7)(9,11,15,17)(10,16,19,23)(12,18,24,20)(13,21,14,22);(9,12,14,23,15,24,13,16)(10,17,18,22,19,11,20,21);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,18,19,20)(11,21,17,22)(12,16,24,23)Polycyclic presentation:
2^8|b^2=h,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245917, #245924, #245941, … |
| maximal normal subgroup of | #245917, #245924, #245941, … |
| maximal quotient of | #293884, #293894, #293904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18614);