Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2084 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,18,23,14)(8,25,27,33)(9,28)(10,16)(11,30)(12,17)(13,20,32,21)(15,26)(19,31)(22,35,24,34)(29,36);(5,7,11,22,12,23,10,24)(6,13,16,33,17,32,30,25)(8,26,20,19,27,29,21,9)(14,15,35,28,18,36,34,31);(5,8,11,20,12,27,10,21)(6,14,16,35,17,18,30,34)(7,19,22,29,23,9,24,26)(13,31,33,15,32,28,25,36)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250508, #72250516, #72250540, … |
| maximal normal subgroup of | #72250508, #72250516, #72250540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56526);