Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,20,22,24);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,20,23,21,22,24,18);(1,4,5,3)(2,7,8,6)(17,20,21,24)(18,22,23,19)Polycyclic presentation:
2^8|a^2=egh,b^2=eg,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100756, #100768, #100877, … |
| maximal normal subgroup of | #100756, #100768, #100877, … |
| maximal quotient of | #362285, #362928, #365670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22214);