Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 | |
2636 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)(4,6)(5,8)(9,10)(11,22)(12,27)(13,20)(14,19)(15,32)(16,21)(17,25)(18,29)(23,37)(24,40)(26,35)(28,39)(30,38)(31,34)(33,36);(1,3)(2,5)(4,8)(6,7)(9,11,14,25)(10,17,19,22)(12,28,24,31)(13,30,23,29)(15,26,16,33)(18,37,38,20)(21,35,32,36)(27,34,40,39);(9,12,14,24)(10,18,19,38)(11,23,25,13)(15,30,16,29)(17,34,22,39)(20,35,37,36)(21,40,32,27)(26,31,33,28);(2,6)(5,7)(10,19)(17,22)(18,38)(20,37)(21,32)(27,40)(34,39)(35,36);(3,8)(5,7)(9,13)(10,20)(11,24)(12,25)(14,23)(15,31)(16,28)(17,27)(18,36)(19,37)(21,39)(22,40)(26,30)(29,33)(32,34)(35,38)Polycyclic presentation:
2^9|b^2=i,c^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574964);