Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
2274 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)(5,6,9,12)(7,8,11,10)(15,19)(18,20)(22,26)(24,31)(27,33)(29,32);(14,17)(18,20)(21,22,25,26)(23,27,30,33)(24,28,31,34)(29,35,32,36);(5,7,9,11)(6,10,12,8)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,34)(32,36);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(21,23,25,30)(22,27,26,33)(24,32,31,29)(28,36,34,35);(5,8,9,10)(6,11,12,7)(13,15)(14,18)(16,19)(17,20)(21,24,25,31)(22,28,26,34)(23,29,30,32)(27,35,33,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4328262);