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) = C26 |
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 | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
741 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);(1,2,3,4);(5,7)(6,11)(8,18)(9,15)(10,16)(12,19)(13,20)(14,17);(6,12)(11,19)(13,17)(14,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105914, #105915, #105920, … |
| maximal normal subgroup of | #105914, #105915, #105920, … |
| maximal quotient of | #188353, #188354, #188355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13642);