Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C22×C4×D4 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,10,5,19)(3,12,14,8,15,23,20,22)(4,9,18,25,16,24,17,29)(13,26,28,21,27,32,30,31);(1,3)(2,8)(4,13)(5,20)(6,14)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,30)(18,28)(24,31)(25,32)(26,29);(1,4,6,18,7,16,5,17)(2,9,11,25,10,24,19,29)(3,13,14,28,15,27,20,30)(8,21,23,32,22,31,12,26);(4,16)(9,24)(13,27)(17,18)(21,31)(25,29)(26,32)(28,30);(2,10)(8,22)(9,24)(11,19)(12,23)(21,31)(25,29)(26,32)Polycyclic presentation:
2^8|a^2=g,c^2=g,f^2=h,g^2=h,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #364596, #364599, #364602, … |
| maximal normal subgroup of | #364596, #364599, #364602, … |
| maximal quotient of | #6344525, #6344526, #6344527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53170);