Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 425 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(11,15)(13,16)(17,18,20,23,19,21,24,22);(2,5)(6,7)(9,10,12,14)(11,16,15,13)(18,21)(22,23);(9,11,12,15)(10,13,14,16)(18,22)(20,24)(21,23);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=g,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #102751, #102752, #102758, … |
| maximal normal subgroup of | #102751, #102752, #102758, … |
| maximal quotient of | #362892, #363829, #368593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22342);