Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(15,22)(16,19)(20,27)(21,25)(23,26);(6,9)(10,12)(13,14,19,25,17,21,16,18)(15,20,26,28,23,27,22,24);(5,7,8,11)(6,10,9,12)(13,15,16,22,17,23,19,26)(14,20,18,24,21,27,25,28);(2,4)(14,21)(18,25)(20,27)(24,28)Polycyclic presentation:
2^8|b^2=fh,c^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6472862, #6472863, #6472866, … |
| maximal normal subgroup of | #6472862, #6472863, #6472866, … |
| maximal quotient of | #350922, #350923, #350928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26035);