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) = C25 |
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×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,6,10,16,24)(3,8,12,22,14,23,27,32)(5,7,15,19,17,20,28,30)(11,21,25,31,13,18,26,29);(2,7)(4,15)(8,18)(10,20)(12,26)(16,28)(21,23)(25,27);(1,3,5,13)(2,8,7,18)(4,12,15,26)(6,14,17,11)(9,22,19,29)(10,23,20,21)(16,27,28,25)(24,32,30,31);(3,11)(8,21)(12,25)(13,14)(18,23)(22,31)(26,27)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dgh55 conjugacy classes — view the full character table.
| maximal subgroup of | #480861, #480864, #480867, … |
| maximal normal subgroup of | #480861, #480864, #480867, … |
| maximal quotient of | #164332, #164335, #164358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10410);