Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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 | |
3312 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,13)(14,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,28)(31,32);(1,2)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,19,22,29)(18,23,25,26)(20,27,24,31)(21,28,30,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,22,24)(18,21,25,30)(19,27,29,31)(23,28,26,32);(1,4)(2,7)(3,10)(5,13)(6,14)(8,15)(9,12)(11,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,25)(24,30)(26,29)(27,28)(31,32);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,21)(18,24)(19,28)(20,25)(22,30)(23,31)(26,27)(29,32)Polycyclic presentation:
2^8|b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #347444, #347464, #347490, … |
| maximal normal subgroup of | #347444, #347464, #347490, … |
| maximal quotient of | #7627411, #7627413, #7627446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56042);