Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
495 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)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,24,20)(25,26,28,30)(27,29,31,32);(10,11)(12,16)(18,22)(20,23)(25,26,28,30)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,23,22,20)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,23)(18,24,22,19)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^8|a^2=eg,b^2=efh,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16145, #610243, #610277, … |
| maximal normal subgroup of | C2×#16145, #610243, #610277, … |
| maximal quotient of | #154839, #154875, #154890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9133);