Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
255 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,8)(7,15,19,16)(9,14,10,13)(12,18,20,17)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(2,4)(5,7,11,19)(6,12,8,20)(9,18,10,17)(13,16,14,15);(5,8,11,6)(7,16,19,15)(9,13,10,14)(12,17,20,18)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eh,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #93798, #93836, #125964, … |
| maximal normal subgroup of | #93798, #93836, #125964, … |
| maximal quotient of | #102462, #102467, #102573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3267);