Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×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 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,7,8)(10,11)(13,14,18,16)(15,20,19,17);(2,4)(9,10,12,11)(13,15)(14,17)(16,20)(18,19);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)(13,16,18,14)(15,20,19,17);(13,17,18,20)(14,19,16,15)Polycyclic presentation:
2^8|a^2=efg,b^2=eh,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #395724, #395960, #395961, … |
| maximal normal subgroup of | #395724, #395960, #395961, … |
| maximal quotient of | #154187, #154388, #154404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8225);