Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 65 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 — 3 generators on 24 points:
(1,2)(3,4)(10,17)(11,22)(12,14)(13,24)(15,18)(19,20)(21,23);(5,6)(7,8)(9,10)(11,12)(13,23)(14,17)(15,22)(16,21)(18,20)(19,24);(2,4)(6,8)(9,11,12,10,18,17,13,22,16,19,24,21,15,23,14,20)Polycyclic presentation:
2^8|c^2=d,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^b=dg,e^b=egh,f^b=fgh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93848, #93850, #151282, … |
| maximal normal subgroup of | #93848, #93850, #151282, … |
| maximal quotient of | #145367, #145383, #145456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6436);