Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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×#3285 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,23)(16,28)(17,33)(18,35)(19,22)(20,26)(21,30)(24,34)(25,41)(27,40)(29,38)(31,36)(32,39)(37,43)(42,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,27,34,17)(14,20,24,23,22,33,35,40)(16,29,32,31,30,25,42,43)(21,37,39,41,28,36,44,38);(6,9)(10,11)(13,16)(14,21)(15,25)(17,31)(18,32)(19,30)(20,36)(22,28)(23,38)(24,39)(26,43)(27,29)(33,37)(34,42)(35,44)(40,41);(2,4)(5,7,8,12)(6,10,9,11)(14,22)(16,30)(20,33)(23,40)(24,35)(25,29)(31,43)(32,42);(13,17,34,27,19,26,18,15)(14,23,35,20,22,40,24,33)(16,31,42,29,30,43,32,25)(21,38,44,36,28,41,39,37)Polycyclic presentation:
2^9|b^2=f,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7458211);