Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 3 generators on 24 points:
(1,2,3,4)(10,14)(12,19)(15,23)(16,20)(17,21)(22,24);(5,6,7,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23);(9,11,13,18)(10,15,16,24)(12,21,19,17)(14,22,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94084, #94088, #146629, … |
| maximal normal subgroup of | #94084, #94088, #146629, … |
| maximal quotient of | #105766, #105768, #105769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3738);