Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2560 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)(11,20)(12,17)(13,29)(14,18)(15,19)(16,22)(21,37)(23,32)(24,36)(25,27)(26,40)(28,35)(30,34)(31,38)(33,39);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,25)(13,18)(14,29)(15,19)(16,23)(17,27)(21,31)(22,32)(24,36)(26,33)(28,34)(30,35)(37,38)(39,40);(9,11,15,24)(10,16,19,32)(12,26,28,21)(13,23,14,22)(17,33,35,31)(18,36,29,20)(25,40,34,37)(27,38,30,39);(2,6)(5,7);(3,8)(5,7)(9,12,15,28)(10,17,19,35)(11,21,24,26)(13,27,14,30)(16,31,32,33)(18,34,29,25)(20,37,36,40)(22,38,23,39)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^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, 6575607);