Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
3180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,9)(7,11)(10,15)(14,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(1,2,6,5)(3,7,9,11)(4,8,12,13)(10,14,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31);(1,5,6,2)(3,11,9,7)(4,8,12,13)(10,14,15,16);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh73 conjugacy classes — view the full character table.
| maximal subgroup of | #6352603, #6352633, #6352663, … |
| maximal normal subgroup of | #6352603, #6352633, #6352663, … |
| maximal quotient of | #7626604, #7626636, #7626689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55916);