Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13469 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2065 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2065 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 153 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:
(1,2,9,13,8,14,7,27)(3,15,4,19,18,28,23,30)(5,24,26,32,6,25,20,12)(10,22,31,16,29,21,11,17);(1,3,5,16,7,4,20,21,8,18,6,17,9,23,26,22)(2,10,12,15,13,11,24,30,14,29,32,28,27,31,25,19);(1,4)(2,11)(3,7)(5,17)(6,16)(8,23)(9,18)(10,13)(12,28)(14,31)(15,32)(19,25)(20,21)(22,26)(24,30)(27,29)Polycyclic presentation:
2^8|a^2=de,b^2=d,d^2=g,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,e^b=eh,f^b=fh,d^c=dg,e^c=egh,f^c=fgh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13469, #499310, #499514, … |
| maximal normal subgroup of | C2×#13469, #499310, #499514, … |
| maximal quotient of | #145909, #145919, #145951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6457);