Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,15)(7,12,18,27)(8,13,22,30)(9,11,23,14)(16,28,19,24)(17,26,32,35)(20,29,21,25)(31,36,33,34);(6,11)(7,16)(8,20)(9,23)(14,15)(18,19)(21,22)(24,28)(25,29)(26,34)(31,33)(35,36);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,24)(13,26)(14,28)(15,27)(16,23)(20,31)(21,33)(22,32)(25,34)(29,36)(30,35);(5,8)(6,13)(7,17)(9,21)(10,22)(11,25)(12,26)(14,29)(15,30)(16,31)(18,32)(19,33)(20,23)(24,34)(27,35)(28,36);(7,18)(12,27)(16,19)(17,32)(24,28)(26,35)(31,33)(34,36)Polycyclic presentation:
2^9|a^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349807);