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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 61 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)(7,17)(8,13)(9,15)(10,26)(11,16)(12,19)(14,23)(18,27)(20,32)(21,34)(22,33)(24,30)(25,31)(28,36)(29,35);(2,4)(5,7,11,21)(6,12,16,29)(8,22,25,36)(9,18,10,20)(13,17,31,34)(14,32,30,27)(15,33,26,28)(19,24,35,23);(6,13)(7,18)(8,23)(9,10)(12,27)(14,15)(16,31)(17,33)(19,35)(20,21)(24,25)(26,30)(28,34)(29,32)Polycyclic presentation:
2^8|b^2=h,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13332, #498859, #498860, … |
| maximal normal subgroup of | C2×#13332, #498859, #498860, … |
| maximal quotient of | #142633, #142634, #142637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6320);