Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | |
2622 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,13)(14,16)(15,20)(18,28)(19,34)(21,36)(22,27)(23,25)(24,37)(26,32)(29,31)(30,33)(35,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,22)(14,32)(15,24)(16,26)(18,36)(19,23)(20,37)(21,28)(25,34)(29,38)(30,40)(31,35)(33,39);(9,12)(10,18)(11,21)(13,20)(14,30)(15,33)(16,29)(17,35)(19,31)(22,32)(23,40)(24,27)(25,38)(26,39)(28,34)(36,37);(1,4,5,3)(2,6,7,8)(9,13)(10,12)(11,22)(14,29)(15,30)(16,31)(17,27)(18,34)(19,28)(20,33)(21,23)(24,40)(25,36)(26,35)(32,38)(37,39);(9,14)(10,16)(11,23)(12,18)(13,28)(15,20)(17,25)(19,34)(21,36)(22,38)(24,32)(26,37)(27,35)(29,30)(31,33)(39,40)Polycyclic presentation:
2^9|b^2=f,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483938);