Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | |
2129 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)(9,10,13,17)(11,20,23,21)(12,18,26,33)(14,28,15,19)(16,29,31,24)(22,35,38,36)(25,27,34,30)(32,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,21)(12,25)(13,17)(14,19)(15,28)(16,24)(18,30)(20,23)(22,37)(26,34)(27,33)(29,31)(32,38)(35,40)(36,39);(9,11,15,29,13,23,14,24)(10,16,28,21,17,31,19,20)(12,22,30,40,26,38,27,39)(18,32,34,36,33,37,25,35);(2,6)(5,7)(10,17)(12,26)(16,31)(19,28)(20,21)(22,38)(27,30)(39,40);(3,8)(5,7)(9,12)(10,18)(11,22)(13,26)(14,27)(15,30)(16,32)(17,33)(19,25)(20,35)(21,36)(23,38)(24,39)(28,34)(29,40)(31,37)Polycyclic presentation:
2^9|a^2=i,c^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589857);