Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C2×D4 · Out(G) = C210 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(6,9)(10,12)(13,15,17,16)(14,18,20,19);(2,4)(14,19)(15,16)(18,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|b^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #208696, #208702, #208708, … |
| maximal normal subgroup of | #208696, #208702, #208708, … |
| maximal quotient of | #241409, #241413, #241421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16557);