Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2236 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)(13,14,18,17,19,23,20,24)(15,25,27,38,28,36,29,21)(16,30,34,40,33,22,35,32)(26,39,43,37,42,41,31,44);(2,4)(13,15,19,28)(14,21,23,38)(16,31,33,43)(17,36,24,25)(18,29,20,27)(22,39,30,41)(26,34,42,35)(32,44,40,37);(6,9)(10,12)(13,16,18,34,19,33,20,35)(14,22,17,32,23,30,24,40)(15,26,27,43,28,42,29,31)(21,37,25,41,38,44,36,39);(13,17,18,23,19,24,20,14)(15,21,27,25,28,38,29,36)(16,32,34,30,33,40,35,22)(26,37,43,41,42,44,31,39);(5,7,8,11)(6,10,9,12)(16,33)(22,30)(26,42)(31,43)(32,40)(34,35)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374731);