Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,15,14)(11,21,23,18)(12,24,27,33)(13,19,30,32)(16,25,17,20)(22,37,38,28)(26,39,40,35)(29,34,36,31);(2,6)(7,8)(9,11)(10,18)(12,25)(13,22)(14,21)(15,23)(16,24)(17,33)(19,28)(20,27)(26,29)(30,38)(31,39)(32,37)(34,35)(36,40);(9,12,15,27)(10,11,14,23)(13,28,30,37)(16,21,17,18)(19,35,32,39)(20,33,25,24)(22,36,38,29)(26,34,40,31);(1,3,5,4)(2,7,6,8)(9,13)(10,19)(11,22)(12,26)(14,32)(15,30)(16,31)(17,34)(18,28)(20,36)(21,37)(23,38)(24,39)(25,29)(27,40)(33,35);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,12,23,27)(13,29,30,36)(16,25,17,20)(18,33,21,24)(19,34,32,31)(22,37,38,28)(26,35,40,39)Polycyclic presentation:
2^9|a^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,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, 6485475);