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 |
| prime radical | C2×#41 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,13)(14,19)(15,21)(17,28)(18,30)(20,32)(22,35)(23,27)(24,33)(25,37)(26,34)(29,31)(36,38)(39,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,26)(17,35)(18,36)(19,33)(20,37)(21,34)(22,28)(25,32)(29,40)(30,38)(31,39);(9,12,14,28,15,29,32,18)(10,17,20,13,21,30,19,31)(11,22,25,27,26,36,24,40)(16,23,33,35,34,39,37,38);(9,13,14,30,15,31,32,17)(10,18,20,12,21,28,19,29)(11,23,25,35,26,39,24,38)(16,22,33,27,34,36,37,40);(1,4,5,3)(2,7,8,6)(9,14,15,32)(10,19,21,20)(11,24,26,25)(12,18,29,28)(13,30,31,17)(16,33,34,37)(22,27,36,40)(23,38,39,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=fi,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, 7470783);