Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C43 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,12,9,14,20,28,10,13,21,27,22,29,35,36)(7,15,17,31,18,24,33,26,16,30,19,34,32,11,23,25);(5,7)(6,11,13,24)(8,19)(9,18)(10,16)(12,26,27,25)(14,15,29,30)(17,21)(20,23)(22,32)(28,34,36,31)(33,35);(1,2,3,4)(7,16)(11,24)(15,30)(17,19)(18,32)(23,33)(25,26)(31,34)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,b^a=bg,e^a=eh,c^b=ch,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93441, #94014, #126529, … |
| maximal normal subgroup of | #93441, #94014, #126529, … |
| maximal quotient of | #126524, C8×#347, #126534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5529);