Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,21)(11,18,25,34)(12,17,14,20)(13,27,30,36)(16,22,32,24)(19,26,37,29)(23,35,28,31)(33,38,40,39);(1,3,5,2)(4,6,8,7)(9,11,14,24,15,25,12,22)(10,16,17,18,21,32,20,34)(13,28,29,38,30,23,26,39)(19,35,27,40,37,31,36,33);(1,4,5,8)(2,6,3,7)(9,12,15,14)(10,17,21,20)(11,22,25,24)(13,26,30,29)(16,18,32,34)(19,27,37,36)(23,38,28,39)(31,33,35,40);(2,3)(6,7)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,20)(17,32)(19,33)(21,34)(26,38)(27,35)(28,30)(29,39)(31,36)(37,40);(9,13,14,29,15,30,12,26)(10,19,17,27,21,37,20,36)(11,23,24,39,25,28,22,38)(16,31,18,33,32,35,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,e^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7453129);