Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| 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 | #60458 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,17)(14,19)(15,20)(18,25)(21,31)(22,36)(23,32)(26,39)(27,33)(28,37)(29,34)(30,35)(38,40);(1,3,4,8)(2,5,6,7)(9,11,15,23)(10,16,20,32)(12,14,22,25)(13,27,30,26)(17,33,35,39)(18,24,19,36)(21,28,38,29)(31,37,40,34);(9,12,25,23,15,22,14,11)(10,16,19,36,20,32,18,24)(13,26,29,38,30,27,28,21)(17,31,37,33,35,40,34,39);(2,6)(5,7);(3,8)(5,7)(9,13)(10,17)(11,21)(12,26)(14,28)(15,30)(16,31)(18,34)(19,37)(20,35)(22,27)(23,38)(24,39)(25,29)(32,40)(33,36)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,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,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, 6590444);