Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
601 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 601 conjugacy classes of subgroups — too many to draw. Only its 175 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)(7,17)(8,22,24,13)(9,16)(10,14)(11,15)(12,19)(18,32,34,31)(20,27)(21,28)(23,29,25,30)(26,33,36,35);(2,4)(5,7,10,20,11,21,9,19)(6,12,16,28,15,27,14,17)(8,18,25,35,24,34,23,33)(13,26,30,32,22,36,29,31);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,35)(21,34)(27,36)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93508, #93509, #93511, … |
| maximal normal subgroup of | #93508, #93509, #93511, … |
| maximal quotient of | #105562, #105563, #105566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3747);