Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| 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 | |
3360 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28);(3,9)(7,13)(10,15)(14,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28);(2,6)(7,13)(8,11)(14,16)(18,25)(19,30)(20,27)(28,32);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #511849, #511852, #10580603, … |
| maximal normal subgroup of | #511849, #511852, #10580603, … |
| maximal quotient of | #7626372, #7626385, #7626413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55873);