Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | #17086 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2376 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)(7,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33);(6,9)(10,11)(13,15,17,20)(14,16,18,19)(23,30)(26,29)(31,36)(33,35);(5,7,8,12)(6,10,9,11)(13,14,17,18)(15,19,20,16)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)(21,23,25,30)(22,26,28,29)(24,31,32,36)(27,33,34,35);(5,7)(6,10)(8,12)(9,11)(13,14)(15,16)(17,18)(19,20)(21,24,25,32)(22,27,28,34)(23,31,30,36)(26,33,29,35)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4506200);