Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2211 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)(7,11)(10,12)(13,14)(15,24)(16,22)(17,18)(19,33)(20,23)(21,26)(25,41)(27,37)(28,44)(29,30)(31,40)(32,35)(34,38)(36,43)(39,42);(6,9)(7,11)(13,15,19,26,20,27,18,34)(14,21,17,24,23,38,33,37)(16,28,30,42,31,25,35,43)(22,39,32,44,40,36,29,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,31)(14,22,23,40)(15,25,27,28)(17,32,33,29)(18,35,19,30)(21,36,38,39)(24,41,37,44)(26,43,34,42);(6,9)(10,12)(13,17,20,33)(14,18,23,19)(15,21,27,38)(16,29,31,32)(22,30,40,35)(24,26,37,34)(25,39,28,36)(41,42,44,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,17,23,33)(15,26,27,34)(16,30,31,35)(21,37,38,24)(22,29,40,32)(25,42,28,43)(36,41,39,44)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050969);