Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2088 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,13,16,24)(7,17,21,19)(8,22)(9,23)(10,11)(14,29)(15,32,25,28)(18,31)(20,30,26,34)(27,36);(1,2)(3,4)(5,6,8,13,11,15,22,28,12,16,9,24,10,25,23,32)(7,18,19,35,20,14,30,27,21,29,17,33,26,31,34,36);(2,4)(5,7,8,19,11,20,22,30,12,21,9,17,10,26,23,34)(6,14,13,27,15,29,28,33,16,31,24,36,25,18,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250243, #72250248, #72250256, … |
| maximal normal subgroup of | #72250243, #72250248, #72250256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56082);