Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461251 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,16,30,15,19,14,13)(11,20,31,17,24,22,23,34)(12,25,27,39,28,18,32,29)(21,35,37,36,26,38,40,33);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,26,28,21)(13,22)(14,31)(15,24)(16,23)(18,33,25,36)(19,34)(20,30)(27,37,32,40)(29,38,39,35);(3,7)(5,8)(9,12,14,27,15,28,16,32)(10,18,13,29,19,25,30,39)(11,21,23,37,24,26,31,40)(17,33,20,35,34,36,22,38);(9,13,16,19,15,30,14,10)(11,22,23,34,24,20,31,17)(12,18,32,29,28,25,27,39)(21,36,37,35,26,33,40,38)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,c^2=e,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378356);