Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
885 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 885 conjugacy classes of subgroups — too many to draw. Only its 271 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(11,15)(13,16)(18,21)(19,23);(1,2,3,4)(5,7,8,6);(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,21)(23,24);(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=fg,d^2=f,b^a=be,c^a=cf,d^a=dg88 conjugacy classes — view the full character table.
| Fitting subgroup of | #14711⋊C3 |
| maximal subgroup of | #119714, #135486, #135490, … |
| maximal normal subgroup of | #119714, #135486, #135490, … |
| maximal quotient of | #154084, #154089, #154090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7699);