Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2072 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)(14,17)(15,19)(21,22,26,29)(23,24,31,27)(25,28,34,36)(30,32,35,33);(6,9)(10,12)(14,17)(18,20)(21,23,26,31)(22,27,29,24)(25,30,34,35)(28,33,36,32);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,36)(34,35);(1,2)(3,4)(21,24)(22,23)(25,33)(26,27)(28,35)(29,31)(30,36)(32,34);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,33)(29,36)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1198624);