Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
412 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)(7,8)(9,11)(10,12)(13,14,16,18)(15,22,23,28)(17,19,25,21)(20,24,27,26);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,18)(15,23)(19,21)(24,26);(2,4)(5,8,9,10)(6,7,11,12)(14,19)(17,25)(18,21)(20,28)(22,27)(24,26);(6,11)(7,12)(13,15,16,23)(14,20,18,27)(17,26,25,24)(19,28,21,22)Polycyclic presentation:
2^8|a^2=h,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632653, #6632662, #6632686, … |
| maximal normal subgroup of | #6632653, #6632662, #6632686, … |
| maximal quotient of | #335676, #335678, #335684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23588);