Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1791 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)(9,10,15,18,16,19,14,33)(11,20,22,32,24,31,23,13)(12,25,28,36,29,35,34,17)(21,30,39,40,26,27,38,37);(1,2)(3,7)(4,6)(5,8)(9,11,14,23,16,24,15,22)(10,13,18,20,19,32,33,31)(12,26,28,38,29,21,34,39)(17,27,35,40,36,30,25,37);(1,3)(2,5)(4,8)(6,7)(9,12,16,29)(10,17,19,36)(11,21,24,26)(13,30,32,27)(14,28,15,34)(18,35,33,25)(20,37,31,40)(22,38,23,39);(3,8)(5,7)(9,13)(10,11)(12,27)(14,31)(15,20)(16,32)(17,26)(18,22)(19,24)(21,36)(23,33)(25,38)(28,37)(29,30)(34,40)(35,39);(2,6)(5,7)(9,14,16,15)(10,18,19,33)(11,22,24,23)(12,28,29,34)(13,31,32,20)(17,35,36,25)(21,38,26,39)(27,37,30,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910260);