Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,15,12,16,10,30)(7,17,19,29,20,28,14,9)(8,21,24,13,26,33,25,32)(18,23,31,36,22,27,34,35);(1,2)(3,4)(5,7,9,16,10,19,28,30,12,20,29,6,11,14,17,15)(8,22,23,32,25,34,35,21,26,18,27,13,24,31,36,33);(2,4)(5,8)(6,13)(7,18)(9,27)(10,25)(11,24)(12,26)(14,34)(15,33)(16,32)(17,35)(19,31)(20,22)(21,30)(23,29)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250231, #72250243, #72250252, … |
| maximal normal subgroup of | #72250231, #72250243, #72250252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55509);