Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 153 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:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,19,21)(20,22,24,23);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,21)(22,23);(10,14)(13,16)(18,21)(22,23);(1,4,5,3)(2,6,7,8)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=e,b^2=ef,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #100381, #100384, #100413, … |
| maximal normal subgroup of | #100381, #100384, #100413, … |
| maximal quotient of | #392394, #392736, #392875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26135);