Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2494 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)(5,6,9,11)(7,8,12,10)(15,19)(17,20)(21,22,25,26)(23,27,30,32)(24,28,31,33)(29,34,35,36);(5,7)(6,8)(9,12)(10,11)(13,14,16,18)(15,17,19,20)(22,26)(24,31)(27,32)(29,35);(5,7,9,12)(6,10,11,8)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,8)(6,7)(9,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848456);