Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1984 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)(7,8)(11,12)(13,14,17,20)(15,19,16,18)(21,22)(23,28)(24,29)(25,30)(26,27)(31,36)(32,33)(34,35);(6,10)(11,12)(13,14,17,20)(15,18,16,19)(22,27)(24,28)(30,35)(32,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,19)(17,20)(21,23,26,29)(22,28,27,24)(25,33,34,31)(30,32,35,36);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15,17,16)(14,18,20,19)(21,24,26,28)(22,29,27,23)(25,32,34,36)(30,33,35,31);(5,8,9,7)(6,12,10,11)(13,16,17,15)(14,19,20,18)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,36)(29,33)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293540);