Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,4,7,6)(9,10,15,24,13,12,14,18)(11,19,23,17,21,16,22,20);(1,3,5,7)(2,6,8,4)(9,11,13,21)(10,16,12,19)(14,22,15,23)(17,24,20,18);(1,4,5,6)(2,7,8,3)(9,12,13,10)(11,20,21,17)(14,18,15,24)(16,22,19,23);(2,8)(3,7)(9,11)(10,17)(12,20)(13,21)(14,23)(15,22)(16,18)(19,24)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125730, #126215, #126387, … |
| maximal normal subgroup of | #125730, #126215, #126387, … |
| maximal quotient of | #345112, #345117, #345120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24219);