Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = 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 | |
501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,24,23)(26,28)(30,31);(2,5)(6,8)(17,18)(19,21)(20,22)(23,24)(25,26,29,28)(27,31,32,30);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24)(25,27,29,32)(26,30,28,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,24)(22,23)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^8|a^2=f,b^2=efgh,c^2=g,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17000, #611884, #611899, … |
| maximal normal subgroup of | C2×#17000, #611884, #611899, … |
| maximal quotient of | #155759, #155763, #155820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9988);