Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 | |
715 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 715 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)(6,7)(11,15)(13,16)(17,18,21,23)(19,20,22,24);(5,6,8,7)(9,10,12,14)(11,13,15,16)(17,19)(18,20)(21,22)(23,24);(1,2,3,4)(5,7,8,6)(17,20)(18,22)(19,23)(21,24);(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^8|a^2=g,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 | #104067, #104099, #104131, … |
| maximal normal subgroup of | #104067, #104099, #104131, … |
| maximal quotient of | #154189, #154405, #154422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8228);