Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
757 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 757 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(2,3)(6,8)(9,10,15,20)(11,16,22,17)(12,23,21,24)(13,19,14,18);(1,2,4,3)(9,11,15,22)(10,16,20,17)(12,13,21,14)(18,23,19,24);(5,6,7,8)(9,12,14,11,15,21,13,22)(10,17,19,24,20,16,18,23);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=fh,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138909, #250904, #250984, … |
| maximal normal subgroup of | #138909, #250904, #250984, … |
| maximal quotient of | #297358, #297367, #297376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18748);