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 | #3500 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1707 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)(11,20)(12,26)(13,14)(15,19)(16,24)(17,32)(18,28)(21,38)(22,25)(23,34)(27,37)(29,31)(30,36)(33,40)(35,39);(2,6)(7,8)(9,11,15,24)(10,16,19,20)(12,27,31,40)(13,22,18,34)(14,23,28,25)(17,35,36,38)(21,30,39,32)(26,33,29,37);(1,3,5,4)(2,7,6,8)(9,12,28,30,15,31,14,32)(10,17,13,29,19,36,18,26)(11,21,23,40,24,39,25,27)(16,33,34,38,20,37,22,35);(1,3,5,4)(2,7,6,8)(9,13,28,19,15,18,14,10)(11,22,23,16,24,34,25,20)(12,29,30,36,31,26,32,17)(21,35,40,33,39,38,27,37);(1,4,5,3)(2,7,6,8)(9,14,15,28)(10,18,19,13)(11,23,24,25)(12,30,31,32)(16,34,20,22)(17,29,36,26)(21,27,39,40)(33,35,37,38)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,e^2=f,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, 7472630);