Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,20)(8,27)(9,16)(10,17)(11,30)(12,19)(13,18)(14,24)(15,29)(21,32)(22,34)(23,31)(25,36)(26,33)(28,35);(5,7)(6,14)(8,28)(9,22)(10,21)(11,26)(12,25)(13,24)(15,32)(16,27)(17,35)(18,36)(19,33)(20,30)(23,29)(31,34);(2,4)(5,8)(6,15)(7,21)(9,25)(10,11)(12,29)(13,22)(14,31)(16,20)(17,18)(19,27)(23,26)(24,28)(30,34)(32,36)(33,35);(5,9,12,21)(6,16,19,32)(7,22,25,10)(8,26,29,24)(11,23,13,28)(14,27,33,15)(17,20,34,36)(18,35,30,31);(5,10,12,22)(6,17,19,34)(7,23,25,28)(8,11,29,13)(9,24,21,26)(14,32,33,16)(15,18,27,30)(20,31,36,35)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480837);