Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| 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 | #106 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#106 |
| 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)(7,18,21,31)(8,14)(9,26)(10,27)(11,28)(12,17)(13,20,30,22)(15,25)(16,24)(19,32,23,36)(29,34,33,35);(2,4)(5,7,10,22,11,23,24,34,12,21,9,20,8,19,25,35)(6,13,16,31,14,29,27,36,17,30,15,18,28,33,26,32);(5,8,12,11)(6,14,17,28)(7,19,21,23)(9,24,10,25)(13,29,30,33)(15,26,16,27)(18,32,31,36)(20,34,22,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146335, #146345, #146431, … |
| maximal normal subgroup of | #146335, #146345, #146431, … |
| maximal quotient of | #126653, #126654, #126690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5551);