Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | #60600 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 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)(4,8)(5,6)(9,10)(11,19)(12,24)(13,28)(14,32)(15,18)(16,23)(17,34)(20,21)(22,36)(25,38)(26,31)(27,30)(29,40)(33,37)(35,39);(2,6)(7,8)(9,11,15,23)(10,16,18,19)(12,25,27,39)(13,29,31,21)(14,33,34,22)(17,37,32,36)(20,26,40,28)(24,35,30,38);(9,12)(10,13)(11,20)(14,26)(15,27)(16,35)(17,30)(18,31)(19,38)(21,36)(22,39)(23,40)(24,32)(25,33)(28,34)(29,37);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,21)(14,30)(15,31)(16,25)(17,26)(18,27)(19,39)(20,36)(22,38)(23,29)(24,34)(28,32)(33,35)(37,40);(1,4,5,3)(2,7,6,8)(9,14,15,34)(10,17,18,32)(11,22,23,33)(12,26,27,28)(13,30,31,24)(16,36,19,37)(20,39,40,25)(21,38,29,35)Polycyclic presentation:
2^9|b^2=i,d^2=fi,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=cg,d^b=dh,e^b=ei,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, 7428469);