Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13545 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2101 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,11,15,14)(7,17,18,32)(8,12)(10,21)(13,28)(16,25,23,24)(19,34,33,26)(20,27)(30,31)(35,36);(1,2)(3,4)(5,6,12,25,21,23,8,14,9,15,27,24,10,16,20,11)(7,13,26,36,33,28,17,29,18,31,34,35,19,30,32,22);(2,4)(5,7,8,17,10,19,12,26,9,18,20,32,21,33,27,34)(6,13,11,22,16,30,24,35,15,31,14,29,23,28,25,36)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72240845, #72240849, … |
| maximal normal subgroup of | #72239851, #72240845, #72240849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56202);