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 | |
3273 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)(17,18,22,23)(19,24,25,30)(20,21,28,29)(26,27,31,32);(3,10)(4,12)(7,9)(8,13)(18,23)(20,28)(24,30)(26,31);(2,6)(7,9)(8,13)(14,15)(19,25)(24,30)(26,31)(27,32);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,24,23,30)(20,26,28,31)(21,27,29,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,21)(19,26)(22,28)(23,29)(24,27)(25,31)(30,32);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,20)(19,27)(22,29)(23,28)(24,26)(25,32)(30,31)Polycyclic presentation:
2^8|a^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 | #347036, #347420, #347422, … |
| maximal normal subgroup of | #347036, #347420, #347422, … |
| maximal quotient of | #7627619, #7627620, #7627622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55962);