Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1868 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)(6,8)(7,11)(15,19)(18,20)(21,22,26,30)(23,24,28,27)(25,29,34,36)(31,32,33,35);(6,8)(10,12)(14,17)(18,20)(21,23,26,28)(22,27,30,24)(25,31,34,33)(29,35,36,32);(5,6,9,8)(7,12,11,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,33)(26,28)(27,30)(29,35)(31,34)(32,36);(1,2)(3,4)(5,7,9,11)(6,10,8,12)(21,24)(22,28)(23,30)(25,32)(26,27)(29,33)(31,36)(34,35);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,33)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5127262);