Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,13,19,9)(8,21)(10,27)(11,17)(12,16)(14,24)(15,28,20,29)(18,31,34,36)(22,23,35,30)(25,32)(26,33);(1,2)(3,4)(5,7,10,15,12,19,11,20)(6,13,17,29,16,9,27,28)(8,22,24,18,26,35,25,34)(14,31,21,23,32,36,33,30);(2,4)(5,8,12,26)(6,14,16,32)(7,18,19,34)(9,23,13,30)(10,25,11,24)(15,22,20,35)(17,33,27,21)(28,31,29,36);(5,9)(6,15)(7,17)(8,23)(10,29)(11,28)(12,13)(14,22)(16,20)(18,33)(19,27)(21,34)(24,31)(25,36)(26,30)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal quotient of | #72238860, #72238861 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291081);