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) = 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 | |
709 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)(6,8)(11,15)(14,16)(18,20)(22,23);(5,6)(7,8)(10,13)(14,16)(17,18,21,20)(19,22,24,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23);(1,2,3,4)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|b^2=ef,c^2=ef,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #258021, #258022, #258029, … |
| maximal normal subgroup of | #258021, #258022, #258029, … |
| maximal quotient of | #154184, #154409, #154410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8162);