Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2034 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,15,16,31)(8,19)(9,23)(10,13)(11,17,26,32)(12,21)(14,25)(18,33,24,28)(20,27,35,30)(22,34)(29,36);(1,2)(3,4)(11,26)(15,31)(27,30)(28,33);(1,3,4,2)(5,7,9,24,10,16,25,18)(6,11,14,30,13,26,23,27)(8,20,21,17,22,35,36,32)(12,28,19,15,29,33,34,31)Polycyclic presentation:
2^9|a^2=g,c^2=dh,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^c=di,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249879, #72249885, #72249897, … |
| maximal normal subgroup of | #72249879, #72249885, #72249897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55827);