Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3496 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,22,23,24);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,23)(11,18)(14,22)(17,24)(19,20);(1,4,5,3)(2,7,8,6)(10,16)(14,17)(15,23)(22,24)Polycyclic presentation:
2^8|b^2=e,c^2=ef,d^2=eh,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245618, #245622, #245630, … |
| maximal normal subgroup of | #245618, #245622, #245630, … |
| maximal quotient of | #303097, #303098, #303099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18889);