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 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,8)(9,24)(10,14)(11,16)(12,17)(13,29)(15,21)(18,26)(19,23)(20,32)(22,30)(25,35)(27,33)(28,34)(31,36);(5,7)(6,10)(8,16)(9,18)(11,30)(12,19)(13,25)(14,21)(15,23)(17,22)(20,34)(24,36)(26,33)(27,35)(28,31)(29,32);(2,4)(5,8)(6,7)(9,25)(10,21)(11,22)(12,23)(13,31)(14,15)(16,30)(17,19)(18,34)(20,27)(24,35)(26,28)(29,36)(32,33);(5,9)(6,13)(7,18)(8,20)(10,25)(11,27)(12,29)(14,31)(15,33)(16,34)(17,24)(19,32)(21,28)(22,36)(23,26)(30,35);(5,10,12,22)(6,14,17,30)(7,15,19,16)(8,21,23,11)(9,26,29,20)(13,32,24,18)(25,28,36,27)(31,33,35,34)Polycyclic presentation:
2^9|e^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, 6480853);