Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2071 |
| 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 | |
424 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)(5,6,12,19)(7,21,24,8)(9,28,10,18)(11,16,32,17)(13,22,30,20)(14,33,34,23)(15,31,29,26)(25,27,35,36);(5,7,13,35,12,24,30,25)(6,14,20,31,19,34,22,26)(8,17,36,18,21,16,27,28)(9,29,11,33,10,15,32,23);(5,8,9,26,30,36,32,14,12,21,10,31,13,27,11,34)(6,15,16,7,22,23,18,35,19,29,17,24,20,33,28,25)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250059, #72250063, #72250068, … |
| maximal normal subgroup of | #72250059, #72250063, #72250068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56330);