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) = C24×C4 |
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 | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,10,22)(3,13)(4,18)(5,19)(9,24,25,32)(11,23,12,20)(15,29)(16,30)(17,28)(21,26,31,27);(1,2)(3,9)(4,8)(5,20)(6,11)(7,12)(10,14)(13,27)(15,21)(16,24)(17,26)(18,23)(19,22)(25,28)(29,31)(30,32);(1,3,14,28)(2,9,10,25)(4,15,19,29)(5,16,18,30)(6,17,7,13)(8,21,22,31)(11,26,12,27)(20,24,23,32);(1,4,7,5)(2,8,12,20)(3,15,13,16)(6,18,14,19)(9,21,27,24)(10,22,11,23)(17,30,28,29)(25,31,26,32)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=h,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 | #106043, #106191, #106197, … |
| maximal normal subgroup of | #106043, #106191, #106197, … |
| maximal quotient of | #160583, #160608, #160609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10581);