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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1885 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)(3,7)(4,8)(5,6)(9,10)(11,23)(12,19)(13,20)(14,21)(15,18)(16,22)(17,25)(24,39)(26,33)(27,35)(28,32)(29,37)(30,34)(31,38)(36,40);(2,6)(7,8)(9,11,14,25)(10,17,21,23)(12,15,26,16)(13,27,29,40)(18,19,22,33)(20,36,37,35)(24,31,28,30)(32,38,39,34);(10,18)(11,12)(15,16)(19,33)(20,34)(21,22)(24,27)(25,26)(28,40)(30,31)(35,36)(37,38);(1,3,5,4)(2,7,6,8)(9,12,14,26)(10,19,21,33)(11,16,25,15)(13,28,29,24)(17,18,23,22)(20,32,37,39)(27,31,40,30)(34,35,38,36);(1,4,5,3)(2,8,6,7)(9,13,14,29)(10,20,21,37)(11,24,25,28)(12,27,26,40)(15,30,16,31)(17,32,23,39)(18,34,22,38)(19,35,33,36)Polycyclic presentation:
2^9|b^2=i,d^2=f,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, 6485231);