Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| 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 | #31072 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,15,18,16,19,14,32)(11,13,22,29,23,31,21,30)(12,24,27,36,28,35,33,17)(20,37,38,40,39,34,25,26);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,25)(14,21)(15,22)(16,23)(17,34)(18,29)(19,31)(20,27)(24,26)(28,38)(30,32)(33,39)(35,40)(36,37);(9,12,16,28)(10,17,19,36)(11,20,23,39)(13,26,31,40)(14,27,15,33)(18,35,32,24)(21,38,22,25)(29,34,30,37);(2,7)(3,8)(9,13)(10,11)(12,26)(14,29)(15,30)(16,31)(17,20)(18,21)(19,23)(22,32)(24,25)(27,34)(28,40)(33,37)(35,38)(36,39);(1,4,5,6)(2,3,7,8)(9,14,16,15)(10,18,19,32)(11,21,23,22)(12,27,28,33)(13,29,31,30)(17,35,36,24)(20,38,39,25)(26,34,40,37)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411244);