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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2580 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,24)(13,30)(14,20)(15,32)(16,21)(18,28)(19,31)(22,35)(23,34)(25,33)(26,36)(27,37)(29,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11,16,13)(10,17,21,30)(12,25,29,39)(14,23,15,31)(18,35,37,36)(19,20,34,32)(22,27,26,28)(24,33,38,40);(9,12)(10,18)(11,22)(13,26)(14,28)(15,27)(16,29)(17,33)(19,36)(20,38)(21,37)(23,39)(24,32)(25,31)(30,40)(34,35);(9,13,16,11)(10,19,21,34)(12,26,29,22)(14,31,15,23)(17,32,30,20)(18,36,37,35)(24,40,38,33)(25,27,39,28);(2,6)(5,7)(9,14,16,15)(10,20,21,32)(11,23,13,31)(12,27,29,28)(17,34,30,19)(18,24,37,38)(22,25,26,39)(33,36,40,35)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=i,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,g^b=gi,e^c=ei,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, 7421541);