Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21);(6,8)(11,16)(14,15)(17,19,20,24)(18,21,22,23);(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,4)(9,12,13,10)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=ef,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 | C2×#14819, #607763, #607764, … |
| maximal normal subgroup of | C2×#14819, #607763, #607764, … |
| maximal quotient of | #154104, #154225, #154232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7807);