Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,28)(25,38)(26,39)(27,34)(29,35)(30,36)(31,37)(33,40)(41,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,28,19,26,17,27)(14,20,22,24,23,34,32,39)(16,25,30,40,29,41,31,42)(21,33,37,38,35,44,36,43);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,30)(19,29)(20,33)(22,37)(23,35)(24,38)(26,41)(27,42)(28,40)(32,36)(34,44)(39,43);(5,7,8,11)(6,10,9,12)(15,26)(16,29)(20,34)(21,35)(24,39)(27,28)(30,31)(36,37);(6,9)(10,12)(13,17,19,18)(14,22,23,32)(15,27,26,28)(16,30,29,31)(20,24,34,39)(21,36,35,37)(25,40,41,42)(33,43,44,38)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427484);