Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501052 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,17,27,14,20,30,33,16,12,13,19,15,21,29,28)(11,18,32,38,31,35,39,37,25,26,23,34,24,22,36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,18)(12,26)(13,23)(14,31)(15,24)(16,25)(17,32)(19,34)(20,35)(21,22)(27,38)(28,40)(29,36)(30,39)(33,37);(9,12,16,10)(11,22,25,35)(13,28,17,33)(14,20,15,21)(18,31,26,24)(19,30,27,29)(23,38,32,34)(36,40,39,37);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fi,b^2=e,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408515);