Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | |
1909 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)(12,26)(13,18)(14,20)(15,21)(17,28)(19,33)(22,35)(23,24)(25,31)(27,38)(29,36)(30,40)(32,37)(34,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,27)(13,15)(14,24)(17,35)(18,21)(19,31)(20,23)(22,28)(25,33)(26,38)(29,39)(30,32)(34,36)(37,40);(9,12,15,29)(10,17,21,37)(11,22,13,30)(14,32,33,28)(16,34,18,38)(19,36,20,26)(23,35,31,40)(24,27,25,39);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,18,21,16)(12,27,29,39)(14,25,33,24)(17,35,37,40)(19,23,20,31)(22,32,30,28)(26,38,36,34);(1,4,5,3)(2,6,8,7)(9,10)(11,23)(12,17)(13,31)(14,19)(15,21)(16,24)(18,25)(20,33)(22,35)(26,28)(27,34)(29,37)(30,40)(32,36)(38,39)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485866);