Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #502133 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24803 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,13,27,15,26,28,25,12,18,9,16,11,8,10,17)(7,14,30,35,33,36,21,19,23,32,20,31,22,24,29,34);(2,4)(5,7,9,20,15,33,10,29,12,23,13,30,11,22,28,21)(6,14,16,31,26,36,17,34,18,32,27,35,8,24,25,19);(5,8,12,26)(6,15,18,11)(7,19,23,34)(9,27,13,16)(10,17,28,25)(14,29,32,21)(20,24,30,36)(22,31,33,35);(2,4)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409335);