Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,12,15)(7,16,18,9)(8,10,14,11)(13,19,20,17);(5,7)(6,13)(8,9)(10,19)(11,17)(12,18)(14,16)(15,20);(5,8,11,6,12,14,10,15)(7,16,17,20,18,9,19,13);(5,9)(6,7)(8,19)(10,20)(11,13)(12,16)(14,17)(15,18)Polycyclic presentation:
2^8|a^2=g,c^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^b=de,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22693, #6380173, #6380174, … |
| maximal normal subgroup of | C2×#22693, #6380173, #6380174, … |
| maximal quotient of | #225498, #225503, #225508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15681);