Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 524288 |
| 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 | |
655 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,19)(20,24,21,23);(2,4)(5,6,7,8)(10,11)(14,15)(17,19,22,18)(20,24,21,23);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,20,22,21)(18,23,19,24);(9,12,13,16)(10,14,11,15)(17,21,22,20)(18,24,19,23)Polycyclic presentation:
2^8|a^2=fg,b^2=eh,c^2=f,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 | #104064, #104078, #104096, … |
| maximal normal subgroup of | #104064, #104078, #104096, … |
| maximal quotient of | #154189, #154395, #154403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8230);