Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2499 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)(7,11)(10,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,27)(21,31)(26,34)(28,36)(29,35)(32,37)(33,38)(39,40)(41,44)(42,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,27,19,29,18,28)(14,20,23,25,24,36,22,35)(16,26,33,42,32,41,31,40)(21,34,37,39,38,44,30,43);(6,9)(10,12)(13,16,18,31,19,32,17,33)(14,21,22,30,24,38,23,37)(15,26,28,40,29,41,27,42)(20,34,35,43,36,44,25,39);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,37,38,30)(26,40,41,42)(34,39,44,43);(5,7,8,11)(6,10,9,12)(16,32)(21,38)(26,41)(30,37)(31,33)(34,44)(39,43)(40,42)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401583);