Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,24,20)(27,31)(29,32);(2,5)(3,7)(11,16)(13,15)(18,22)(20,23)(25,26,28,30)(27,32,31,29);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,27,28,31)(26,29,30,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,20,21,23)(18,24,22,19)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=efg,b^2=efh,c^2=g,d^2=efg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| Fitting subgroup of | #17308⋊C3 |
| maximal subgroup of | #125830, #126019, #126157, … |
| maximal normal subgroup of | #125830, #126019, #126157, … |
| maximal quotient of | #156640, #156673, #156683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10296);