Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,17,22)(15,23,16,19)(18,27,26,21)(20,24,28,25);(1,2)(3,4)(6,9)(7,12)(13,15,17,16)(14,19,22,23)(18,24,26,25)(20,21,28,27);(2,4)(13,16,17,15)(14,20,22,28)(18,24,26,25)(19,21,23,27);(5,7,8,12)(6,10,9,11)(14,21)(18,26)(19,28)(20,23)(22,27)(24,25)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632671, #6632678, #6632687, … |
| maximal normal subgroup of | #6632671, #6632678, #6632687, … |
| maximal quotient of | #340704, #340706, #340712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23917);