Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
453 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)(5,6,9,13,10,12,8,17)(7,14,15,11,16,20,18,19);(5,7,9,15,10,16,8,18)(6,11,13,20,12,19,17,14);(1,2,3,4);(6,12)(11,19)(13,17)(14,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #109635, #109636, #109760, … |
| maximal normal subgroup of | #109635, #109636, #109760, … |
| maximal quotient of | #203616, #203617, #203618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14165);