Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,18,28)(8,12)(9,14)(10,13)(11,17,24,21)(16,29,31,36)(19,26)(20,25)(22,27)(23,30,34,32)(33,35);(1,2,3,4)(5,7)(6,11)(8,16)(9,21)(10,18)(12,23)(13,24)(14,28)(15,27)(17,22)(19,32)(20,31)(25,34)(26,36)(29,35)(30,33);(5,8,9,19,10,20,22,33)(6,12,14,26,13,25,27,35)(7,16,21,32,18,31,17,30)(11,23,28,36,24,34,15,29);(6,13)(11,24)(12,25)(14,27)(15,28)(23,34)(26,35)(29,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,b^a=be,d^a=dh,e^b=eh,f^b=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #135270, #146976, #146977, … |
| maximal normal subgroup of | #135270, #146976, #146977, … |
| maximal quotient of | #183336, #183339, #183344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13432);