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 | |
1970 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)(11,23)(12,27)(13,18)(14,20)(15,21)(16,33)(17,22)(19,30)(24,37)(25,39)(26,35)(28,36)(29,40)(31,38)(32,34);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,18,21,23)(12,14,29,32)(16,26,17,28)(19,37,39,38)(20,27,34,40)(22,35,33,36)(24,30,31,25);(1,3,5,4)(2,7,8,6)(9,12)(10,19)(11,24)(13,31)(14,28)(15,29)(16,25)(17,30)(18,34)(20,23)(21,39)(22,40)(26,32)(27,33)(35,37)(36,38);(1,4,5,3)(2,6,8,7)(9,13,15,11)(10,18,21,23)(12,14,29,32)(16,28,17,26)(19,38,39,37)(20,40,34,27)(22,35,33,36)(24,30,31,25);(9,14,15,32)(10,20,21,34)(11,25,13,30)(12,28,29,26)(16,31,17,24)(18,19,23,39)(22,37,33,38)(27,36,40,35)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=f,e^2=i,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=ehi,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, 6484795);