Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=g,e^2=h,f^2=g,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 | #106049, #106192, #107265, … |
| maximal normal subgroup of | #106049, #106192, #107265, … |
| maximal quotient of | #160580, #160594, #160599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10662);