Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60647 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,12,7)(8,19,24,32)(9,25,27,31)(10,15,11,18)(13,29,16,22)(14,26,17,21)(20,33,30,36)(23,34,28,35);(1,2,4,3)(5,7,12,6)(8,20,24,30)(9,17,27,14)(10,15,11,18)(13,23,16,28)(19,33,32,36)(21,31,26,25)(22,35,29,34);(5,8,21,33,11,23,27,29,12,24,26,36,10,28,9,22)(6,13,25,35,15,20,17,32,7,16,31,34,18,30,14,19);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410323);