Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500112 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(7,15)(8,18)(9,22)(11,24)(13,28)(14,27)(16,32)(17,20)(19,34)(21,35)(23,33)(25,31)(26,29)(30,36);(1,2)(3,4)(5,6)(7,11)(8,19)(9,14)(10,12)(13,21)(15,31)(16,25)(17,26)(18,34)(20,36)(22,27)(23,30)(24,32)(28,35)(29,33);(2,4)(5,7,9,23,10,16,22,17)(6,11,14,30,12,25,27,26)(8,20,18,32,21,33,35,15)(13,29,28,31,19,36,34,24);(6,12)(8,21)(11,25)(14,27)(15,32)(18,35)(20,33)(26,30)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fhi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407308);