Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(11,21)(12,26)(13,28)(14,32)(15,20)(16,25)(17,29)(18,31)(19,34)(22,30)(23,40)(24,37)(27,39)(33,38)(35,36);(2,6)(7,8)(9,11,15,25)(10,16,20,21)(12,27,18,36)(13,30,17,23)(14,33,34,24)(19,38,32,37)(22,28,40,29)(26,35,31,39);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,17,20,13)(11,22,25,40)(14,28,34,29)(16,35,21,39)(19,31,32,26)(23,37,30,38)(24,36,33,27);(9,13,15,17)(10,18,20,12)(11,23,25,30)(14,26,34,31)(16,36,21,27)(19,29,32,28)(22,37,40,38)(24,35,33,39);(1,4,5,3)(2,7,6,8)(9,14,15,34)(10,19,20,32)(11,24,25,33)(12,28,18,29)(13,26,17,31)(16,37,21,38)(22,36,40,27)(23,35,30,39)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,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,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7427221);