Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2256 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)(6,9)(10,12)(15,19)(18,20)(21,22,26,27)(23,28,29,24)(25,33,34,30)(31,32,36,35);(14,17)(18,20)(22,27)(23,29)(25,34)(32,35);(1,2)(3,4)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,33)(34,36);(5,6)(7,10)(8,9)(11,12)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=fg,c^2=fh,d^2=f,e^2=i,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, 4271327);