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 |
| prime radical | #25436 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 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)(11,23)(12,18)(13,19)(14,29)(15,21)(16,22)(17,26)(20,30)(24,33)(25,35)(27,37)(28,32)(31,36)(34,39)(38,40);(1,3,5,2)(4,6,8,7)(9,11,15,26,16,13,12,24)(10,17,18,23,22,33,21,19)(14,28,30,39,31,25,27,40)(20,35,29,34,37,32,36,38);(1,4,5,8)(2,6,3,7)(9,12,16,15)(10,18,22,21)(11,24,13,26)(14,27,31,30)(17,23,33,19)(20,29,37,36)(25,39,28,40)(32,38,35,34);(2,3)(6,7)(9,13)(10,19)(11,16)(12,26)(14,28)(15,24)(17,18)(20,34)(21,33)(22,23)(25,31)(27,40)(29,32)(30,39)(35,36)(37,38);(9,14,12,27,16,31,15,30)(10,20,21,36,22,37,18,29)(11,25,24,39,13,28,26,40)(17,32,19,34,33,35,23,38)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gi,e^2=fi,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, 7453132);