Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,9,4,14)(5,11,6,10)(8,15,16,13)(17,18,22,25,23,19,21,27)(20,26,29,32,30,24,28,31);(1,3,7,4)(2,8,12,16)(5,13,6,15)(9,11,14,10)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(1,4,7,3)(2,9,12,14)(5,13,6,15)(8,11,16,10)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32);(1,5,7,6)(2,10,12,11)(3,13,4,15)(8,9,16,14)(17,21,23,22)(18,25,19,27)(20,28,30,29)(24,31,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dg,e^a=eg,d^b=dh,e^b=eg,d^c=dgh58 conjugacy classes — view the full character table.
| maximal subgroup of | #484045, #484048, #484059, … |
| maximal normal subgroup of | #484045, #484048, #484059, … |
| maximal quotient of | #182573, #183681, #183946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11081);