Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,12)(7,8,11,9)(14,17)(15,19)(21,22)(23,24)(25,33)(26,27)(28,30)(29,35)(31,34)(32,36);(5,7,10,11)(6,8,12,9)(14,17)(18,20)(22,27)(23,30)(29,33)(31,36);(5,8,10,9)(6,7,12,11)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,27)(25,31)(26,30)(29,34)(32,33)(35,36);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,24,26,28)(22,23,27,30)(25,34,35,32)(29,36,33,31);(5,9,10,8)(6,11,12,7)(13,15,16,19)(14,18,17,20)(21,25,26,35)(22,29,27,33)(23,31,30,36)(24,32,28,34)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1167233);