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 | #2099 |
| 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)(5,6,10,16,12,17,29,14,11,18,30,15,28,34,9,27)(7,19,23,32,24,33,26,36,22,35,21,13,8,25,20,31);(2,4)(5,7,12,8,11,22,28,24)(6,13,17,36,18,32,34,31)(9,23,10,20,29,21,30,26)(14,33,15,19,27,25,16,35);(2,4)(5,8,12,22,11,24,28,7)(6,13,17,36,18,32,34,31)(9,26,10,23,29,20,30,21)(14,25,15,35,27,33,16,19)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=eh,g^b=ghi,h^b=hi,d^c=dhi,e^c=ehi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250568, #72250578, #72250598, … |
| maximal normal subgroup of | #72250568, #72250578, #72250598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56569);