Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 32768 |
| 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 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,19,22,38)(12,23,27,37)(13,28,25,17)(15,30,32,20)(16,33,35,26)(21,39,29,34)(24,36,40,31);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11,14,22)(10,15,18,32)(12,24,27,40)(13,29,25,21)(16,34,35,39)(17,36,28,31)(19,37,38,23)(20,33,30,26);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=h,c^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 | #139049, #139055, #142629, … |
| maximal normal subgroup of | #139049, #139055, #142629, … |
| maximal quotient of | #496259, #496670, #496813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26743);