Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #399 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,15,11,16,9,14)(7,12,20,28,18,26,19,27)(8,13,23,31,22,30,24,32)(17,33,21,36,34,25,35,29);(1,2,3,4)(5,7)(6,12)(8,21,22,35)(9,19)(10,20)(11,18)(13,29,30,36)(14,27)(15,28)(16,26)(17,23,34,24)(25,31,33,32);(5,8)(6,13)(7,17)(9,24)(10,23)(11,22)(12,25)(14,32)(15,31)(16,30)(18,34)(19,35)(20,21)(26,33)(27,36)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,e^b=eh,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11084, #357754, #358666, … |
| maximal normal subgroup of | C2×#11084, #357754, #358666, … |
| maximal quotient of | #115974, #115990, #115993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4072);