Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #427 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1808 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,12,7,29)(3,14,18,41,20,40,19,50)(4,21,23,36,24,10,28,35)(5,25,26,37,27,56,15,11)(9,30,33,48,32,42,39,17)(16,44,46,62,47,38,49,61)(22,53,43,52,54,55,57,34)(31,58,59,45,51,63,60,64);(1,3,15,42,7,19,27,48,8,20,26,30,6,18,5,17)(2,9,25,41,13,33,37,40,12,32,56,50,29,39,11,14)(4,16,43,63,23,46,54,64,24,47,57,58,28,49,22,45)(10,31,55,62,36,60,52,44,21,51,53,61,35,59,34,38);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,31)(11,34)(12,21)(13,36)(14,38)(15,43)(17,45)(18,49)(19,46)(20,47)(25,55)(26,57)(27,54)(29,35)(30,58)(32,51)(33,60)(37,52)(39,59)(40,44)(41,62)(42,63)(48,64)(50,61)(53,56)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94218, #94220, #134773, … |
| maximal normal subgroup of | #94218, #94220, #134773, … |
| maximal quotient of | #134675, #134676, #134696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5828);