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 |
| prime radical | #61055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 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)(11,21)(12,19)(13,18)(14,17)(15,20)(16,24)(22,32)(23,37)(25,31)(26,40)(27,34)(28,33)(29,36)(30,39)(35,38);(2,5)(3,7)(9,11,13,24)(10,16,18,21)(12,26,27,38)(14,25,15,22)(17,32,20,31)(19,35,34,40)(23,28,39,29)(30,33,37,36);(1,3,4,7)(2,6,5,8)(10,17)(11,22)(12,27)(14,15)(18,20)(19,36)(23,26)(24,25)(30,37)(31,32)(33,34)(38,39);(2,5)(6,8)(10,18)(16,21)(17,20)(19,34)(30,37)(31,32)(33,36)(35,40);(9,12)(10,19)(11,23)(13,27)(14,28)(15,29)(16,30)(17,33)(18,34)(20,36)(21,37)(22,38)(24,39)(25,26)(31,40)(32,35)Polycyclic presentation:
2^9|b^2=i,c^2=fi,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, 6576260);