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 |
| prime radical | #60430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1902 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)(4,5)(6,8)(9,10)(11,20)(12,24)(13,18)(14,17)(15,22)(16,27)(19,28)(21,38)(23,33)(25,37)(26,36)(29,32)(30,35)(31,40)(34,39);(2,5)(6,7)(9,11,13,22)(10,15,18,20)(12,25,26,40)(14,29,28,23)(16,34,35,38)(17,33,19,32)(21,30,39,27)(24,31,36,37);(9,12,13,26)(10,16,18,35)(11,21,22,39)(14,30,28,27)(15,31,20,37)(17,24,19,36)(23,40,29,25)(32,34,33,38);(10,17)(14,28)(15,32)(16,24)(18,19)(20,33)(23,29)(25,40)(27,30)(31,34)(35,36)(37,38);(1,3,4,8)(2,6,5,7)(10,18)(11,22)(16,35)(17,19)(21,39)(23,29)(24,36)(25,40)Polycyclic presentation:
2^9|b^2=i,c^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=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, 7425781);