Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| 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 | |
3264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(4,11)(7,12)(10,15)(17,18,23,28)(19,29,31,24)(20,27,22,25)(21,26,30,32);(3,9)(7,12)(10,15)(14,16)(17,18)(19,29)(20,25)(21,32)(22,27)(23,28)(24,31)(26,30);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,23,31)(18,24,28,29)(20,30,22,21)(25,32,27,26);(17,20,23,22)(18,25,28,27)(19,30,31,21)(24,32,29,26);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,21)(18,26)(19,20)(22,31)(23,30)(24,25)(27,29)(28,32);(1,4)(2,8)(3,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,20)(18,27,28,25)(19,21,31,30)(24,26,29,32)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #185426, #185432, #185438, … |
| maximal normal subgroup of | #185426, #185432, #185438, … |
| maximal quotient of | #7626217, #7626222, #7626227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55819);