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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,22)(17,23)(18,33)(19,34)(20,25)(21,27)(26,37)(28,40)(29,36)(30,43)(31,39)(32,38)(35,41)(42,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,27,20,28,17,18)(14,21,23,24,25,33,34,40)(16,26,29,41,31,42,32,30)(22,35,38,37,39,43,36,44);(6,10)(7,11)(13,16,20,31)(14,22,25,39)(15,26,28,42)(17,32,19,29)(18,30,27,41)(21,35,33,43)(23,38,34,36)(24,37,40,44);(2,4)(6,10)(9,12)(13,17,20,19)(14,23,25,34)(15,18,28,27)(16,29,31,32)(21,24,33,40)(22,36,39,38)(26,41,42,30)(35,44,43,37);(13,18,17,28,20,27,19,15)(14,24,34,21,25,40,23,33)(16,30,32,42,31,41,29,26)(22,37,36,35,39,44,38,43)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,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, 7434591);