Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | C2×2−1+4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 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)(9,10,14,19)(11,21,13,16)(12,24,27,38)(15,31,30,20)(17,32,37,28)(18,23,34,29)(22,35,26,36)(25,33,40,39);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,16,19,21)(12,25,27,40)(15,29,30,23)(17,35,37,36)(18,31,34,20)(22,28,26,32)(24,39,38,33);(9,12)(10,17)(11,22)(13,26)(14,27)(15,32)(16,33)(18,36)(19,37)(20,38)(21,39)(23,25)(24,31)(28,30)(29,40)(34,35);(9,13,14,11)(10,18,19,34)(12,26,27,22)(15,23,30,29)(16,20,21,31)(17,36,37,35)(24,33,38,39)(25,28,40,32);(1,4,5,3)(2,7,8,6)(10,19)(16,21)(17,37)(18,34)(20,31)(24,38)(33,39)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,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=cgi,d^b=dh,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, 7426505);