Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1860 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)(6,9)(10,12)(15,19)(18,20)(21,22,26,29)(23,30,28,25)(24,32,35,27)(31,34,33,36);(14,17)(18,20)(21,23,26,28)(22,25,29,30)(24,33,35,31)(27,36,32,34);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,31)(26,28)(27,34)(29,30)(32,36)(33,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,24)(22,27)(23,31)(25,34)(26,35)(28,33)(29,32)(30,36);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,28)(23,29)(24,34)(26,30)(27,33)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296181);