Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
449 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)(7,10)(8,9)(11,12)(13,14,19,22,18,24,17,23)(15,21,27,20,16,26,25,28);(1,2)(3,4)(13,15,18,16)(14,20,24,28)(17,27,19,25)(21,23,26,22);(2,4)(6,9)(10,12)(13,16,18,15)(14,21,24,26)(17,27,19,25)(20,23,28,22);(5,7,8,11)(6,10,9,12)(14,22)(17,19)(20,21)(23,24)(25,27)(26,28)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=h,d^2=fh,e^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26631, #6439511, #6439532, … |
| maximal normal subgroup of | C2×#26631, #6439511, #6439532, … |
| maximal quotient of | #310062, #310064, #310070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19619);