Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3277 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,29)(17,22)(18,23)(19,24)(21,25)(26,37)(27,35)(28,36)(30,44)(31,38)(32,40)(33,41)(34,42)(39,43);(6,9)(10,12)(13,15,19,28)(14,20,24,36)(16,18,25,26)(17,31,33,30)(21,37,29,23)(22,38,41,44)(27,34,43,32)(35,42,39,40);(2,4)(5,7,8,11)(6,10,9,12)(15,25)(16,28)(18,26)(20,21)(23,37)(27,31)(29,36)(30,43)(32,34)(35,38)(39,44)(40,42);(13,16,19,25)(14,21,24,29)(15,26,28,18)(17,27,33,43)(20,23,36,37)(22,39,41,35)(30,34,31,32)(38,42,44,40);(13,17,19,33)(14,22,24,41)(15,27,28,43)(16,30,25,31)(18,34,26,32)(20,35,36,39)(21,38,29,44)(23,42,37,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482257);