Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(2,5)(3,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,6)(5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^8|a^2=f,c^2=gh,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105889, #106050, #107399, … |
| maximal normal subgroup of | #105889, #106050, #107399, … |
| maximal quotient of | #160572, #160592, #160595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10571);