Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,23)(13,21)(14,29)(15,19)(17,28)(18,25)(20,34)(22,33)(24,32)(26,37)(27,38)(30,31)(35,36)(39,40);(2,6)(7,8)(9,11,15,22)(10,16,19,33)(12,24,27,39)(13,29,30,18)(14,31,25,21)(17,34,35,26)(20,36,37,28)(23,32,38,40);(1,3,5,4)(2,7,6,8)(9,12,25,36,15,27,14,28)(10,17,29,38,19,35,18,23)(11,20,31,39,22,37,21,24)(13,26,33,40,30,34,16,32);(1,4,5,3)(2,8,6,7)(9,13,14,16,15,30,25,33)(10,11,18,21,19,22,29,31)(12,26,28,32,27,34,36,40)(17,20,23,24,35,37,38,39);(2,6)(7,8)(12,27)(17,35)(20,37)(23,38)(24,39)(26,34)(28,36)(32,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468991);