Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#388 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,14,26,15,8)(4,17,20,32,19,9,18,24)(13,27,30,22,28,25,29,31);(1,3)(2,8,11,23)(4,13)(5,15)(6,16)(7,14)(9,22,17,31)(10,12,21,26)(18,29)(19,28)(20,30)(24,25,32,27);(1,4,5,18,7,19,6,20)(2,9,10,24,11,17,21,32)(3,13,15,29,14,28,16,30)(8,22,12,25,23,31,26,27);(3,14)(8,23)(12,26)(13,28)(15,16)(22,31)(25,27)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #360288, #360289, #360321, … |
| maximal normal subgroup of | #360288, #360289, #360321, … |
| maximal quotient of | #202479, #202480, #202481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14068);