Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 763 conjugacy classes of subgroups — too many to draw. Only its 151 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)(3,4)(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23);(2,4)(9,11,12,18)(10,14,15,22)(13,17,20,19)(16,24,23,21);(6,8)(10,15)(11,17)(13,20)(14,21)(18,19)(22,24);(2,4)Polycyclic presentation:
2^8|a^2=fh,b^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #267399, #267403, #267427, … |
| maximal normal subgroup of | #267399, #267403, #267427, … |
| maximal quotient of | #267069, #267070, #267071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17345);