Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,7)(8,10,12,11)(14,17)(15,19)(21,22,26,30)(23,29,25,27)(24,28,32,35)(31,36,33,34);(6,7)(10,11)(15,19)(18,20)(21,23,26,25)(22,27,30,29)(24,31,32,33)(28,34,35,36);(5,7,9,6)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,36)(30,35);(1,2)(3,4)(13,15,16,19)(14,18,17,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36);(5,8,9,12)(6,10,7,11)(21,25,26,23)(22,29,30,27)(24,31,32,33)(28,34,35,36)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=h,d^2=g,e^2=f,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, 4506902);