Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| 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 | |
3271 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31);(3,10)(4,12)(7,9)(8,13)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32);(2,6)(7,9)(8,13)(14,15)(19,25)(22,29)(26,31)(30,32);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)Polycyclic presentation:
2^8|b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh67 conjugacy classes — view the full character table.
| maximal subgroup of | #347037, #347421, #347423, … |
| maximal normal subgroup of | #347037, #347421, #347423, … |
| maximal quotient of | #7627619, #7627620, #7627621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55963);