Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
638 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)(7,16)(8,21)(9,25)(10,15)(11,18)(12,24)(13,30)(14,23)(17,32)(19,27)(20,36)(22,31)(26,34)(28,35)(29,33);(5,7,8,18,23,17,30,33,10,19,22,35,9,20,24,34)(6,11,13,27,25,26,21,32,15,28,12,16,14,29,31,36);(7,17,19,20)(8,22)(11,26,28,29)(12,13)(16,32,27,36)(18,34,35,33)(21,31)(24,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=hi,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi,c^b=ch,g^b=gi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240739, #72249861, #72249869, … |
| maximal normal subgroup of | #72240739, #72249861, #72249869, … |
| maximal quotient of | #72240741, #72280067, C2×#146346 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53542);