Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| 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 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 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)(11,18)(12,25)(13,19)(14,20)(15,29)(16,21)(17,23)(22,33)(24,32)(26,40)(27,35)(28,37)(30,36)(31,38)(34,39);(1,3,5,4)(2,6,7,8)(9,11,14,23)(10,17,20,18)(12,15,24,16)(13,26,27,39)(19,34,35,40)(21,32,29,25)(22,30,38,28)(31,36,33,37);(9,12,15,11,14,24,16,23)(10,18,21,25,20,17,29,32)(13,26,28,22,27,39,30,38)(19,33,36,40,35,31,37,34);(1,4,5,3)(2,6,7,8)(9,13,14,27)(10,19,20,35)(11,22,23,38)(12,26,24,39)(15,28,16,30)(17,31,18,33)(21,36,29,37)(25,40,32,34);(11,23)(12,24)(17,18)(22,38)(25,32)(26,39)(31,33)(34,40)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351167);