Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | |
2001 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)(11,16)(12,18)(13,17)(14,19)(15,20)(21,30)(22,29)(23,31)(24,32)(25,34)(26,33)(27,36)(28,35)(37,40)(38,39);(2,5)(3,7)(9,11,15,24)(10,16,20,32)(12,25,27,37)(13,23,14,21)(17,31,19,30)(18,34,36,40)(22,28,38,26)(29,35,39,33);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(13,14)(18,33)(19,20)(22,25)(23,24)(26,28)(30,31)(34,40)(35,36)(37,38);(2,5)(6,8);(9,12)(10,18)(11,22)(13,26)(14,28)(15,27)(16,29)(17,33)(19,35)(20,36)(21,25)(23,37)(24,38)(30,34)(31,40)(32,39)Polycyclic presentation:
2^9|b^2=i,c^2=f,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,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576246);