Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 32768 |
| 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 | C23 | C22×D4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 131 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)(9,10,12,14)(11,15,16,13)(18,21)(22,24);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(18,22)(20,23)(21,24);(10,14)(13,15)(17,18)(19,21)(20,24)(22,23);(1,4,5,3)(2,6,7,8)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=efh,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 | #99697, #99700, #165858, … |
| maximal normal subgroup of | #99697, #99700, #165858, … |
| maximal quotient of | #244439, #244451, #244467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16805);