Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 73 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 — 3 generators on 24 points:
(1,2)(3,4)(10,15)(11,17)(12,19)(13,22)(16,21)(18,20)(23,24);(2,4)(5,6)(7,8)(9,10,14,11)(12,20,23,21)(13,18,22,16)(15,24,17,19);(6,8)(9,11,14,10)(12,21,23,20)(13,16,22,18)(15,19,17,24)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125639, #125641, #125662, … |
| maximal normal subgroup of | #125639, #125641, #125662, … |
| maximal quotient of | #142627, #142705, #142826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6315);