Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19304 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3115 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
341 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(2,6)(3,4)(7,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24);(1,2,5,7)(3,6,4,8);(1,3,5,4)(2,6,7,8)(11,18)(14,23)(17,24)(19,22)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #268301, #268310, #268314, … |
| maximal normal subgroup of | #268301, #268310, #268314, … |
| maximal quotient of | #189126, #197821, #197822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12309);