Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19);(5,7,8,16,9,15,17,14)(6,10,12,20,11,18,19,13);(1,2,3,4);(6,11)(10,18)(12,19)(13,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107412, #107414, #107504, … |
| maximal normal subgroup of | #107412, #107414, #107504, … |
| maximal quotient of | #204486, #204487, #204488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14240);