Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13504 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2077 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2077 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,9,25,7,12,6,13,8,15,23,31,5,14,24,26)(3,16,21,28,17,10,18,11,19,30,32,27,4,20,22,29);(1,3)(2,10)(4,5)(6,18)(7,17)(8,19)(9,21)(11,25)(12,30)(13,27)(14,16)(15,20)(22,24)(23,32)(26,28)(29,31);(1,4,5,3,8,17,7,19)(2,11,12,28,15,29,14,27)(6,22,9,21,24,18,23,32)(10,26,16,25,20,13,30,31)Polycyclic presentation:
2^8|a^2=eg,c^2=d,d^2=h,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,e^c=eg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13504, #499394, #499603, … |
| maximal normal subgroup of | C2×#13504, #499394, #499603, … |
| maximal quotient of | #146043, #146047, #146073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6492);