Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 163 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)(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);(1,2,3,4)(11,18)(14,23)(17,24)(19,22);(6,8)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #268140, #268142, #268143, … |
| maximal normal subgroup of | #268140, #268142, #268143, … |
| maximal quotient of | #351772, #351773, #351776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26172);