Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 389 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,15,12,14);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,18,20,23)(19,21,24,22);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(18,21)(19,24)(22,23);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99673, #102556, #102558, … |
| maximal normal subgroup of | #99673, #102556, #102558, … |
| maximal quotient of | #358729, #358808, #358973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16825);