Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574070 |
| 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 | |
2002 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)(3,8)(4,6)(5,7)(9,10,15,20)(11,23,12,17)(13,18,26,34)(14,19,30,37)(16,22,24,21)(25,36,27,33)(28,35,29,40)(31,39,32,38);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,15)(13,24)(14,29)(16,26)(18,22)(19,35)(20,23)(21,34)(25,32)(27,31)(28,30)(33,38)(36,39)(37,40);(9,12,24,26,15,11,16,13)(10,18,22,23,20,34,21,17)(14,27,32,29,30,25,31,28)(19,35,39,36,37,40,38,33);(1,4,5,3)(2,6,7,8)(9,13,15,26)(10,18,20,34)(11,24,12,16)(14,25,30,27)(17,22,23,21)(19,36,37,33)(28,31,29,32)(35,39,40,38);(9,14)(10,19)(11,25)(12,27)(13,28)(15,30)(16,31)(17,33)(18,35)(20,37)(21,38)(22,39)(23,36)(24,32)(26,29)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^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=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491527);