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 | |
3173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,13)(8,14)(15,16)(17,18)(19,27)(20,21)(22,24)(23,26)(25,32)(28,29)(30,31);(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,32,31);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,20,23,25)(18,21,26,32)(19,28,22,30)(24,31,27,29);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,22,23,19)(18,24,26,27)(20,30,25,28)(21,31,32,29);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=h,e^2=gh,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.
| Fitting subgroup of | #62949⋊C3 |
| maximal subgroup of | #346808, #346832, #346856, … |
| maximal normal subgroup of | #346808, #346832, #346856, … |
| maximal quotient of | #7626618, #7626658, #7626713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55937);