Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,18,20,21)(19,22,24,23);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,19,20,24)(18,22,21,23);(9,12,13,11)(10,14,15,16)(18,21)(19,23)(22,24);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=eh,b^2=f,c^2=fh,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #306513, #306514, #306529, … |
| maximal normal subgroup of | #306513, #306514, #306529, … |
| maximal quotient of | #360914, #363058, #365838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19269);