Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
779 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 779 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:
(2,3)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23);(1,2,4,3)(5,6)(7,8)(10,14)(15,23)(16,21)(17,24);(6,8)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17);(1,3,4,2)(10,14)(15,23)(16,21)(17,24)Polycyclic presentation:
2^8|b^2=eh,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101536, #101537, #293323, … |
| maximal normal subgroup of | #101536, #101537, #293323, … |
| maximal quotient of | #261052, #261062, #261072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17205);