Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
277 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 277 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:
(6,13)(7,19)(8,24)(9,27)(10,11)(14,31)(15,18)(16,25)(17,28)(21,29)(22,26)(23,35)(30,32)(34,36);(1,2)(3,4)(5,6,12,18)(7,20,23,33)(8,25,9,28)(10,17,11,16)(13,27,15,24)(14,22,31,21)(19,32,35,36)(26,34,29,30);(2,4)(5,7)(6,14)(8,26)(9,29)(10,22)(11,21)(12,23)(13,30)(15,34)(16,33)(17,20)(18,31)(19,27)(24,35)(25,36)(28,32)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,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dgh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13337, #498877, #498878, … |
| maximal normal subgroup of | C2×#13337, #498877, #498878, … |
| maximal quotient of | #142634, #142635, #142645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6325);