Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(2,4)(5,6,10,15,11,16,9,27,12,17,8,24,26,31,25,13)(7,18,21,14,22,30,20,32,23,35,19,33,29,36,28,34);(1,2,3,4)(6,13,17,27)(8,25)(9,10)(11,26)(14,30,33,36)(15,16,24,31)(18,34,35,32)(19,28)(20,21)(22,29);(5,7,12,23)(6,14,17,33)(8,20,10,28)(9,19,25,21)(11,29,26,22)(13,30,27,36)(15,18,24,35)(16,34,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13038, #494559, #494561, … |
| maximal normal subgroup of | C2×#13038, #494559, #494561, … |
| maximal quotient of | #136706, #136708, #136740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6026);