Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
721 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 721 conjugacy classes of subgroups — too many to draw. Only its 343 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 — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,13)(9,14)(10,24)(11,15)(12,19)(17,30)(18,31)(20,26)(21,27)(22,29)(23,28)(25,33)(32,36)(34,35);(5,7,10,20,11,19,9,18)(6,12,14,26,15,16,24,31)(8,17,21,32,22,33,23,34)(13,25,28,36,29,30,27,35);(1,3,4,2)(5,8,11,22)(6,13,15,29)(7,17,19,33)(9,23,10,21)(12,25,16,30)(14,28,24,27)(18,34,20,32)(26,36,31,35);(5,9,11,10)(6,14,15,24)(7,18,19,20)(8,21,22,23)(12,26,16,31)(13,27,29,28)(17,32,33,34)(25,35,30,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fh,d^2=h,g^2=h,b^a=be,g^a=gh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #138242, #138311, #138313, … |
| maximal normal subgroup of | #138242, #138311, #138313, … |
| maximal quotient of | #181946, #181948, #181957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13387);