Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 686 conjugacy classes of subgroups — too many to draw. Only its 155 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)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24);(1,2,3,4)(9,10,13,15)(11,14,18,22)(12,16,19,23)(17,21,20,24);(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21);(6,8)(10,15)(14,22)(16,23)(21,24)Polycyclic presentation:
2^8|b^2=eh,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6470284, #6470285, #6470286, … |
| maximal normal subgroup of | #6470284, #6470285, #6470286, … |
| maximal quotient of | #347777, #347780, #347783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25906);