Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | #61056 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2001 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,28)(13,19)(14,20)(15,32)(16,21)(17,26)(18,30)(23,38)(24,29)(25,27)(31,34)(33,37)(35,36)(39,40);(2,5)(6,8)(9,11)(10,17)(12,29)(13,24)(14,26)(15,27)(16,33)(18,38)(19,36)(20,22)(21,25)(23,31)(28,35)(30,39)(32,37)(34,40);(1,3,4,7)(2,6,5,8)(9,12,14,19)(10,18,20,34)(11,23,26,39)(13,32,28,21)(15,31,16,30)(17,35,22,24)(25,38,37,40)(27,36,33,29);(9,13)(10,19)(11,24)(12,20)(14,28)(15,18)(16,34)(17,36)(21,31)(22,29)(23,25)(26,35)(27,38)(30,32)(33,40)(37,39);(9,10)(11,25)(12,18)(13,31)(14,20)(15,21)(16,32)(17,33)(19,34)(22,27)(23,38)(24,36)(26,37)(28,30)(29,35)(39,40)Polycyclic presentation:
2^9|c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482150);