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×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2079 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)(16,21)(17,23)(18,24)(19,22)(25,34)(26,36)(27,37)(28,35)(29,38)(30,41)(31,39)(32,40)(33,42)(43,44);(6,9)(10,12)(13,15,19,28)(14,20,22,35)(16,29,32,26)(17,25,30,43)(18,33,31,27)(21,38,40,36)(23,34,41,44)(24,42,39,37);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,31)(18,30)(19,32)(20,34)(22,40)(23,39)(24,41)(26,27)(28,43)(29,33)(35,44)(36,37)(38,42);(14,22)(16,30)(17,32)(18,31)(20,35)(21,23)(25,26)(27,33)(29,43)(34,38)(36,44)(40,41);(13,17,19,30)(14,23,22,41)(15,26,28,29)(16,31,32,18)(20,36,35,38)(21,39,40,24)(25,27,43,33)(34,37,44,42)Polycyclic presentation:
2^9|b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6482181);