Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | |
2126 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)(3,5)(4,6)(7,8)(9,10,14,31,16,20,15,19)(11,21,24,17,26,34,25,35)(12,27,13,18,29,37,28,38)(22,32,30,40,23,36,39,33);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,23)(13,30)(14,25)(15,24)(16,26)(18,36)(19,34)(20,35)(21,31)(22,29)(27,33)(28,39)(32,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,13,16,29,15,28)(10,18,19,27,20,38,31,37)(11,22,25,39,26,23,24,30)(17,32,34,40,35,36,21,33);(3,8)(5,7)(9,13,14,29,16,28,15,12)(10,18,31,37,20,38,19,27)(11,23,24,39,26,22,25,30)(17,33,34,32,35,40,21,36);(2,6)(5,7)(9,14,16,15)(10,19,20,31)(11,24,26,25)(12,28,29,13)(17,21,35,34)(18,37,38,27)(22,39,23,30)(32,40,36,33)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928570);