Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(7,11)(10,12)(13,14,17,16)(15,18,19,20)(21,22,26,23)(24,29,31,27)(25,28,33,30)(32,35,36,34);(1,2,4,3)(15,19)(18,20)(22,23)(24,31)(28,30)(32,36);(2,3)(5,6,9,8)(7,10,11,12)(13,15,17,19)(14,18,16,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35);(5,7)(6,10)(8,12)(9,11)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,20,19,18)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fhi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2439293);