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 | |
3313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,6)(3,10,5,8)(4,12,14,9)(11,15,13,16)(17,18,22,25)(19,26,27,23)(20,21,24,30)(28,32,31,29);(1,3,7,5)(2,8,6,10)(4,11,14,13)(9,15,12,16)(19,27)(23,26)(28,31)(29,32);(1,3,7,5)(2,8,6,10)(4,11,14,13)(9,15,12,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16);(1,5,7,3)(2,8,6,10)(4,13,14,11)(9,15,12,16)(17,20)(18,21)(19,28)(22,24)(23,29)(25,30)(26,32)(27,31);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,21)(18,24)(19,29)(20,25)(22,30)(23,31)(26,28)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #511855, #6350173, #6350174, … |
| maximal normal subgroup of | #511855, #6350173, #6350174, … |
| maximal quotient of | #7627221, #7627230, #7627239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56018);