Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #427 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1846 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
133 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)(3,14,18,38)(4,20,23,50)(5,25)(6,12)(7,29)(8,13)(9,16,32,45)(10,22,34,24)(11,27)(15,39)(17,37,19,47)(21,51,53,33)(26,35)(28,55)(30,57)(31,44,56,46)(36,42)(40,61)(41,59)(43,60)(48,63)(49,52,64,54)(58,62);(1,3,5,16,7,19,27,46,8,18,28,45,6,17,26,44)(2,9,11,14,12,31,25,37,13,32,35,38,29,56,55,47)(4,15,21,40,24,42,54,62,23,43,53,63,22,41,52,57)(10,30,33,36,20,48,49,39,34,58,51,59,50,61,64,60);(1,4)(2,10,13,34)(3,15)(5,21)(6,22)(7,24)(8,23)(9,30,32,58)(11,33,35,51)(12,20,29,50)(14,36,38,59)(16,40)(17,41)(18,43)(19,42)(25,49,55,64)(26,52)(27,54)(28,53)(31,48,56,61)(37,39,47,60)(44,57)(45,63)(46,62)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152284, #152285, #152286, … |
| maximal normal subgroup of | #152284, #152285, #152286, … |
| maximal quotient of | #135734, #135735, #135753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5899);