Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
4481 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)(8,12)(9,25)(10,24)(11,15)(13,20)(14,27)(17,22)(18,33)(19,21)(23,26)(28,30)(29,36)(31,35)(32,34);(5,7)(6,13)(8,19)(9,18)(10,22)(11,21)(12,20)(14,29)(15,17)(16,24)(23,33)(25,32)(26,36)(27,35)(28,34)(30,31);(2,4)(5,8,11,24)(6,10,15,12)(7,17,21,13)(9,25,27,14)(16,20,19,22)(18,33,35,31)(23,26,30,28)(29,36,32,34);(5,9,11,27)(6,14,15,25)(7,18,21,35)(8,23,24,30)(10,26,12,28)(13,29,17,32)(16,31,19,33)(20,34,22,36);(5,6)(7,19)(8,10)(9,23)(11,15)(12,24)(13,20)(14,28)(16,21)(17,22)(18,29)(25,26)(27,30)(31,34)(32,35)(33,36)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,h^b=hi,d^c=dhi,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, 6480852);