Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #635984 |
| automorphism | ? |
| 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 | |
2272 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)(23,24)(25,33)(26,27)(28,29)(30,34)(31,35)(32,36);(6,9)(10,12)(13,14,16,18)(15,17,20,19)(22,27)(23,29)(30,33)(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)(26,29)(30,35)(32,33)(34,36);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,17,18,19)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 544018);