Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13522 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2088 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2088 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,8,21,19,4,6,10,7,12,23,18,5,11,20,28)(3,13,22,25,16,27,15,29,17,26,31,32,14,9,24,30);(1,3,5,14,7,17,19,16)(2,9,4,13,12,27,11,26)(6,15,8,22,20,24,23,31)(10,25,18,29,28,32,21,30);(1,4,5,12,7,11,19,2)(3,9,14,13,17,27,16,26)(6,10,8,18,20,28,23,21)(15,30,22,25,24,29,31,32)Polycyclic presentation:
2^8|a^2=eg,b^2=d,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,f^b=fh,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13522, #499443, #499514, … |
| maximal normal subgroup of | C2×#13522, #499443, #499514, … |
| maximal quotient of | #146166, #146169, #146181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6510);