Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(10,13)(14,16)(17,19,21,20)(18,22,23,24);(2,4)(17,20,21,19)(18,22,23,24);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | #354521, #354522, #354523, … |
| maximal normal subgroup of | #354521, #354522, #354523, … |
| maximal quotient of | #153736, #153738, C4×#1441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6834);