Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17,19,14)(12,16)(13,15)(18,23,24,21)(20,22);(1,2,4,6)(3,8,7,5)(9,11,13,19)(10,14,15,17)(12,18,20,24)(16,21,22,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=gh,e^2=h,b^a=be,c^a=cgh,e^a=eg,e^b=eh,f^b=fgh64 conjugacy classes — view the full character table.
| maximal subgroup of | #106005, #106011, #106179, … |
| maximal normal subgroup of | #106005, #106011, #106179, … |
| maximal quotient of | #160445, #160469, #160495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10563);