Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2557 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)(7,16)(8,13)(9,25)(10,14)(11,15)(12,19)(17,31)(18,20)(21,23)(22,27)(24,30)(26,29)(28,33)(32,34)(35,36);(5,7,9,19,11,20,10,26)(6,12,14,16,15,29,25,18)(8,17,23,33,24,35,27,34)(13,28,22,31,30,32,21,36);(2,4)(5,6)(7,16)(8,21)(9,14)(10,25)(11,15)(12,26)(13,23)(17,32)(18,20)(19,29)(22,24)(27,30)(28,35)(31,34)(33,36);(5,8)(6,13)(7,17)(9,23)(10,27)(11,24)(12,28)(14,22)(15,30)(16,31)(18,36)(19,33)(20,35)(21,25)(26,34)(29,32);(5,6)(7,18)(8,22)(9,14)(10,25)(11,15)(12,19)(13,27)(16,20)(17,32)(21,24)(23,30)(26,29)(28,35)(31,34)(33,36)Polycyclic presentation:
2^9|b^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,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, 6481273);