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 | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
434 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)(7,8)(9,11)(10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15,16,23)(14,18,19,21)(17,22,25,24)(20,26,28,27);(2,4)(5,8,9,10)(6,7,11,12)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(2,4)(6,11)(7,12)(14,19)(15,23)(20,28)(22,24)Polycyclic presentation:
2^8|b^2=h,c^2=eh,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632316, #6632325, #6632344, … |
| maximal normal subgroup of | #6632316, #6632325, #6632344, … |
| maximal quotient of | #335647, #335648, #335651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23551);