Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,17)(13,16)(14,15)(18,22)(20,23)(21,24);(1,3)(2,5)(4,8)(6,7)(9,11,12,14)(10,15,17,19)(13,20,21,22)(16,18,24,23);(3,8)(5,7)(10,16)(11,14)(13,22)(15,23)(17,24)(18,19)(20,21);(2,6)(5,7)(10,17)(15,19)(16,24)(18,23)Polycyclic presentation:
2^8|b^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #139045, #139051, #488518, … |
| maximal normal subgroup of | #139045, #139051, #488518, … |
| maximal quotient of | #496261, #496263, #496673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26742);