Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
665 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)(5,6,10,13,11,8,9,19)(7,14,18,15,16,20,17,12);(5,7,9,17,11,16,10,18)(6,12,13,14,8,15,19,20);(2,4)(5,8)(6,11)(7,15)(9,19)(10,13)(12,16)(14,18)(17,20);(7,16)(12,15)(14,20)(17,18)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #360322, #360323, #360583, … |
| maximal normal subgroup of | #360322, #360323, #360583, … |
| maximal quotient of | #188781, #188785, #188789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13667);