Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2246 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22,19,23,29,17)(15,24,26,35,27,39,33,20)(16,21,32,38,31,37,40,30)(25,36,28,42,41,44,43,34);(2,4)(13,15,19,27)(14,20,23,35)(16,28,31,43)(17,24,22,39)(18,33,29,26)(21,36,37,44)(25,40,41,32)(30,42,38,34);(6,9)(10,12)(13,16,29,40,19,31,18,32)(14,21,17,30,23,37,22,38)(15,25,33,43,27,41,26,28)(20,34,39,44,35,42,24,36);(13,17,29,23,19,22,18,14)(15,20,33,39,27,35,26,24)(16,30,40,37,31,38,32,21)(25,34,43,44,41,42,28,36);(5,7,8,11)(6,10,9,12)(16,31)(21,37)(25,41)(28,43)(30,38)(32,40)(34,42)(36,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374709);