Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
2705 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)(7,11)(10,12)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,28,30,24)(25,33,35,29)(31,32,36,34);(6,9)(10,12)(13,14,16,18)(15,17,20,19)(22,27)(23,30)(29,33)(31,36);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,27)(25,31)(26,30)(29,34)(32,33)(35,36);(1,2)(3,4)(21,24,26,28)(22,23,27,30)(25,34,35,32)(29,36,33,31);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,25,26,35)(22,29,27,33)(23,31,30,36)(24,32,28,34)Polycyclic presentation:
2^9|a^2=fg,b^2=g,d^2=f,e^2=f,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, 4325135);