Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
504 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)(5,6,10,17,9,16,12,28)(7,19,23,34,22,29,26,32)(8,18,11,27,24,14,25,15)(13,20,31,21,30,35,33,36);(5,7,11,20,10,26,24,36,9,22,25,35,12,23,8,21)(6,13,18,32,28,31,15,19,16,30,14,34,17,33,27,29);(2,4)(5,8,10,11,9,24,12,25)(6,14,17,15,16,18,28,27)(7,20,23,21,22,35,26,36)(13,29,31,32,30,19,33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250623, #72250635, #72250653, … |
| maximal normal subgroup of | #72250623, #72250635, #72250653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54397);