Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,14,10,13,8,12)(7,11,15,18,16,19,17,20);(5,7,9,17,10,16,8,15)(6,11,12,18,13,19,14,20);(2,4);(5,8,10,9)(6,12,13,14)(7,15,16,17)(11,18,19,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #106080, #107196, #107256, … |
| maximal normal subgroup of | #106080, #107196, #107256, … |
| maximal quotient of | #195922, #195923, #195924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13977);