Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
693 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 693 conjugacy classes of subgroups — too many to draw. Only its 215 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)(3,4)(5,6,7,8)(10,14)(12,20)(15,23)(16,21)(17,24)(18,22);(9,10,13,16)(11,17,19,23)(12,21,20,14)(15,22,24,18);(2,4);(9,11,13,19)(10,15,16,24)(12,22,20,18)(14,23,21,17)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138921, #138923, #220248, … |
| maximal normal subgroup of | #138921, #138923, #220248, … |
| maximal quotient of | #220120, #220136, #220152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15428);