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) = C24 |
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 | |
635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,15)(12,13)(14,16)(17,18,22,21)(19,24,20,23);(6,8)(10,12)(14,15)(17,19,22,20)(18,23,21,24);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,23,21,24);(1,2)(3,4)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^8|a^2=ef,b^2=ef,c^2=ef,d^2=e,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×#15367, #608900, #609018, … |
| maximal normal subgroup of | C2×#15367, #608900, #609018, … |
| maximal quotient of | #154699, #154734, #154758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8355);