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) = 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 | |
2276 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)(5,6,10,9)(7,12,11,8)(13,14,16,18)(15,17,19,20)(22,23)(24,31)(28,30)(32,36);(15,19)(17,20)(21,22,26,23)(24,27,31,29)(25,30,33,28)(32,35,36,34);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,34,36,35);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9,10,6)(7,12,11,8)(13,14)(15,17)(16,18)(19,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=f,d^2=fg,e^2=g,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, 4219477);