Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,4)(3,9,12,16)(5,13,15,8)(7,14,10,11)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,4)(7,10)(8,13)(9,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18745, #6348357, #6348359, … |
| maximal normal subgroup of | C2×#18745, #6348357, #6348359, … |
| maximal quotient of | #168146, #168179, #168228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11733);