Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,4)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24);(5,6,7,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(2,4)(6,8)(11,21)(12,22)(15,24)(16,19);(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)Polycyclic presentation:
2^8|b^2=eg,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #264659, #264662, #264665, … |
| maximal normal subgroup of | #264659, #264662, #264665, … |
| maximal quotient of | #318056, #318062, #318068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21217);