Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 | |
733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,24,23,20);(1,2,4,3)(6,8)(10,13)(14,16)(18,22)(19,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,20,22,24);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #104789, #104790, #104791, … |
| maximal normal subgroup of | #104789, #104790, #104791, … |
| maximal quotient of | #154154, #154246, #154282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7999);