Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | #60429 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1896 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)(3,7)(4,5)(6,8)(9,10,14,19)(11,15,22,33)(12,23,26,36)(13,17,29,18)(16,28,35,25)(20,37,39,34)(21,31,27,32)(24,40,38,30);(2,5)(6,7)(9,11,14,22)(10,15,19,33)(12,24,26,38)(13,27,29,21)(16,34,35,37)(17,32,18,31)(20,25,39,28)(23,40,36,30);(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,23)(18,36)(19,35)(21,24)(22,39)(25,29)(27,38)(31,37)(32,34)(33,40);(10,17)(13,29)(15,31)(16,23)(18,19)(21,27)(24,38)(25,28)(30,37)(32,33)(34,40)(35,36);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,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, 7425471);