Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| 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 | |
3308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,12)(8,13)(14,16)(17,18)(19,28)(20,21)(22,30)(23,24)(25,29)(26,32)(27,31);(4,11)(8,13)(9,15)(14,16)(18,24)(19,29)(21,26)(27,30);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,25)(20,30)(21,31)(22,26)(23,29)(24,28)(27,32);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20)(18,26)(19,30)(21,24)(22,25)(23,32)(27,29)(28,31);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,21,23,26)(18,20,24,32)(19,31,29,22)(25,30,28,27);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,22,23,31)(18,27,24,30)(19,26,29,21)(20,25,32,28)Polycyclic presentation:
2^8|d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #62839⋊C3 |
| maximal subgroup of | #511848, #6344205, #6344244, … |
| maximal normal subgroup of | #511848, #6344205, #6344244, … |
| maximal quotient of | #7626237, #7626241, #7626283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55827);