Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#120 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,6,10,15,22)(3,8,12,21,11,20,23,30)(5,7,14,18,16,19,26,29)(13,17,25,28,24,27,31,32);(2,7)(4,14)(8,17)(10,19)(12,25)(15,26)(20,27)(23,31);(1,3,5,13)(2,8,7,17)(4,12,14,25)(6,11,16,24)(9,21,18,28)(10,20,19,27)(15,23,26,31)(22,30,29,32);(3,11)(8,20)(12,23)(13,24)(17,27)(21,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dh55 conjugacy classes — view the full character table.
| maximal subgroup of | #480861, #480864, #480867, … |
| maximal normal subgroup of | #480861, #480864, #480867, … |
| maximal quotient of | #164128, #164131, #164132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10400);