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 | |
3255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(19,26)(23,27)(28,31)(29,32);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,18,22,25)(19,27,26,23)(20,21,24,30)(28,32,31,29);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,22,26)(18,23,25,27)(20,28,24,31)(21,29,30,32);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15);(1,5,7,2)(3,9,4,8)(6,10,14,13)(11,15,12,16)(17,20)(18,21)(19,28)(22,24)(23,29)(25,30)(26,31)(27,32);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21)(18,24)(19,29)(20,25)(22,30)(23,31)(26,32)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #63033⋊C3 |
| maximal subgroup of | #6350199, #6350200, #6350223, … |
| maximal normal subgroup of | #6350199, #6350200, #6350223, … |
| maximal quotient of | #7627229, #7627230, #7627232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56021);