Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,15)(13,22)(14,19)(16,21)(18,23)(20,24);(2,5)(6,7)(9,11)(10,14)(12,19)(13,23)(15,17)(16,18)(20,21)(22,24);(9,10)(11,18,19,24)(12,15)(13,16)(14,23,17,20)(21,22);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,22)(18,24)Polycyclic presentation:
2^8|c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268291, #268296, #268319, … |
| maximal normal subgroup of | #268291, #268296, #268319, … |
| maximal quotient of | #293333, #293338, #293341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18583);