Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 159 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)(7,16)(8,13)(9,23)(10,26)(11,15)(12,18)(14,24)(17,31)(19,29)(20,35)(21,28)(22,30)(25,36)(27,33)(32,34);(5,7,9,19,8,17,22,34,11,21,25,18,10,20,24,33)(6,12,14,28,26,32,36,31,15,29,30,16,13,27,23,35);(5,8,11,10)(6,13,15,26)(7,17,21,20)(9,22,25,24)(12,27,29,32)(14,23,30,36)(16,31,28,35)(18,33,19,34)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,f^2=gh,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146343, #146347, #146434, … |
| maximal normal subgroup of | #146343, #146347, #146434, … |
| maximal quotient of | #126666, #126698, #126717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5557);