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 | |
3265 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)(18,23)(19,27)(20,30)(26,31);(3,9)(7,12)(10,15)(14,16)(17,18,22,23)(19,28,27,24)(20,25,30,32)(21,31,29,26);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,22,27)(18,24,23,28)(20,29,30,21)(25,31,32,26);(17,20)(18,25)(19,29)(21,27)(22,30)(23,32)(24,31)(26,28);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,21,22,29)(18,26,23,31)(19,20,27,30)(24,25,28,32);(1,4)(2,8)(3,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20)(18,25)(19,29)(21,27)(22,30)(23,32)(24,31)(26,28)Polycyclic presentation:
2^8|b^2=g,c^2=h,e^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 | #185269, #185275, #185281, … |
| maximal normal subgroup of | #185269, #185275, #185281, … |
| maximal quotient of | #7626212, #7626224, #7626249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55813);