Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(7,11)(10,12)(15,19)(17,20)(21,22,25,29)(23,27,31,26)(24,32,34,28)(30,35,33,36);(6,9)(10,12)(21,23,25,31)(22,26,29,27)(24,33,34,30)(28,36,32,35);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,36)(32,35)(33,34);(1,3,4,2)(13,15)(14,17)(16,19)(18,20)(21,22)(23,26)(24,28)(25,29)(27,31)(30,35)(32,34)(33,36);(5,7,8,11)(6,10,9,12)(21,24,25,34)(22,28,29,32)(23,30,31,33)(26,35,27,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4238613);