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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 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)(3,6)(4,5)(7,8)(9,10,15,21)(11,24,25,17)(12,27,29,39)(13,19,26,40)(14,31,32,20)(16,23,34,22)(18,35,38,30)(28,36,33,37);(2,5)(6,8)(9,11)(10,17)(12,28)(13,16)(14,30)(15,25)(18,20)(19,22)(21,24)(23,40)(26,34)(27,37)(29,33)(31,38)(32,35)(36,39);(9,12,15,29)(10,18,21,38)(11,14,25,32)(13,28,26,33)(16,35,34,30)(17,36,24,37)(19,31,40,20)(22,39,23,27);(1,3,4,7)(2,6,5,8)(9,13,15,26)(10,19,21,40)(11,16,25,34)(12,14,29,32)(17,22,24,23)(18,37,38,36)(20,27,31,39)(28,30,33,35);(9,14,15,32)(10,20,21,31)(11,12,25,29)(13,30,26,35)(16,33,34,28)(17,27,24,39)(18,19,38,40)(22,37,23,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6482651);