Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,14)(7,17,21,28,19,27,20,12)(8,22,24,30,25,29,23,13)(18,32,35,36,33,26,34,31);(1,2)(3,4)(5,7,9,20,11,19,10,21)(6,12,15,17,16,28,14,27)(8,18,23,34,25,33,24,35)(13,26,22,31,30,32,29,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32);(6,14,16,15)(7,19)(12,17,28,27)(13,29,30,22)(18,33)(20,21)(26,31,32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239830, #72252980, #72256873, … |
| maximal normal subgroup of | #72239830, #72252980, #72256873, … |
| maximal quotient of | #72308364 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283332);