Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,20,35)(8,12,10,14)(9,22,25,32)(13,28,31,18)(15,21,33,23)(19,29,26,34)(24,30,36,27);(6,12)(7,18)(8,10)(9,23)(13,29)(14,16)(15,33)(17,27)(19,24)(20,28)(21,25)(26,36)(30,35)(31,34);(2,4)(5,7,8,19,11,20,10,26)(6,13,14,30,16,31,12,27)(9,24,21,18,25,36,23,28)(15,29,32,17,33,34,22,35)Polycyclic presentation:
2^8|a^2=h,c^2=d,d^2=h,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dh,e^b=eg,f^b=fg,g^b=gh,e^c=egh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13414, #499078, #499079, … |
| maximal normal subgroup of | C2×#13414, #499078, #499079, … |
| maximal quotient of | #144893, #144926, #145012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6402);