Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,17,11)(8,12)(9,14)(10,13)(16,27,29,22)(18,28,31,23)(19,25)(20,24)(21,26)(30,35,36,33)(32,34);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,22)(13,15)(14,23)(19,30)(20,29)(21,31)(24,27)(25,33)(26,28)(32,36)(34,35);(5,8,9,19,10,20,21,32)(6,12,14,25,13,24,26,34)(7,16,18,30,17,29,31,36)(11,22,23,33,15,27,28,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,g^2=h,b^a=bd,d^a=dh,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93414, #128767, #128768, … |
| maximal normal subgroup of | #93414, #128767, #128768, … |
| maximal quotient of | #105300, #105301, #105302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3699);