Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 55 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)(5,6,11,17)(7,18,22,29)(8,12,9,25)(10,27,24,16)(13,19,31,20)(14,26,15,23)(21,28,34,32)(30,36,33,35);(6,12)(7,19)(8,23)(9,26)(10,24)(14,16)(15,27)(17,25)(18,32)(20,22)(21,36)(28,29)(30,33)(34,35);(2,4)(5,7,8,19,24,34,26,36,11,22,9,20,10,21,23,35)(6,13,14,29,27,33,25,32,17,31,15,18,16,30,12,28)Polycyclic presentation:
2^8|a^2=h,c^2=d,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13440, #499158, #499159, … |
| maximal normal subgroup of | C2×#13440, #499158, #499159, … |
| maximal quotient of | #145360, #145361, #145392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6428);