Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C23×C8 |
| socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,9)(10,12)(13,14,16,19,17,15,18,20);(5,6)(7,10)(8,9)(11,12)(13,15,16,20,17,14,18,19);(2,4)(5,7,8,11)(6,10,9,12);(6,9)(10,12)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,g^2=h,b^a=be,c^a=cf,d^b=de100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22900, #6381530, #6381534, … |
| maximal normal subgroup of | C2×#22900, #6381530, #6381534, … |
| maximal quotient of | #228270, #228271, #228272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15888);