Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,25)(7,16,19,28,20,27,18,12)(8,21,24,13,22,31,23,30)(17,29,35,36,33,26,34,32);(1,2,3,4)(5,7,9,18,11,20,10,19)(6,12,14,16,15,28,25,27)(8,17,23,34,22,33,24,35)(13,29,31,36,30,26,21,32);(5,8)(6,13)(7,17)(9,23)(10,24)(11,22)(12,26)(14,31)(15,30)(16,32)(18,34)(19,35)(20,33)(21,25)(27,36)(28,29);(8,22)(13,30)(17,33)(21,31)(23,24)(26,29)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72255707, #72255714, #72255722, … |
| maximal normal subgroup of | #72255707, #72255714, #72255722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264898);