Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,10)(3,11,14,8)(4,13,15,5)(6,9,16,12)(17,18,22,25)(19,26,27,23)(20,21,24,30)(28,32,31,29);(1,3,7,14)(2,8,10,11)(4,12,15,9)(5,6,13,16)(19,27)(23,26)(28,31)(29,32);(1,3,7,14)(2,8,10,11)(4,12,15,9)(5,6,13,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(1,4)(2,5)(3,12)(6,8)(7,15)(9,14)(10,13)(11,16);(1,5)(2,4)(3,6)(7,13)(8,12)(9,11)(10,15)(14,16)(17,20)(18,21)(19,28)(22,24)(23,29)(25,30)(26,32)(27,31);(1,6)(2,9)(3,13)(4,8)(5,14)(7,16)(10,12)(11,15)(17,21)(18,24)(19,29)(20,25)(22,30)(23,31)(26,28)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6350171, #6350172, #6350184, … |
| maximal normal subgroup of | #6350171, #6350172, #6350184, … |
| maximal quotient of | #7627213, #7627215, #7627227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56011);