Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563341 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 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)(11,15)(12,23)(13,17)(14,19)(16,26)(18,28)(20,32)(21,31)(22,35)(24,29)(25,33)(27,30)(34,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,24)(13,27)(14,21)(16,32)(17,30)(18,35)(19,31)(20,26)(22,28)(23,29)(25,37)(33,39)(34,40)(36,38);(9,12,13,26,14,25,28,38)(10,16,18,23,19,34,17,33)(11,20,22,24,21,36,27,37)(15,29,30,32,31,39,35,40);(10,17,19,18)(11,21)(15,30,31,35)(16,33,34,23)(20,36)(22,27)(24,37)(29,32,39,40);(1,4,5,3)(2,7,8,6)(12,25)(16,34)(20,36)(23,33)(24,37)(26,38)(29,39)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=hi,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521313);