Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1896 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22,26,27)(23,30,33,28)(24,25,34,29)(31,35,36,32);(7,11)(10,12)(13,15,16,19)(14,17,18,20)(22,27)(25,29)(28,30)(32,35);(6,9)(10,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,33)(22,28,27,30)(24,31,34,36)(25,32,29,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24,26,34)(22,29,27,25)(23,31,33,36)(28,35,30,32);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,29)(22,24,27,34)(23,32,33,35)(28,31,30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5516063);