Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(15,25)(16,30)(17,18)(20,34)(21,37)(22,23)(26,36)(27,41)(28,29)(31,38)(32,33)(35,42);(2,4)(5,7,9,8)(6,10,12,11)(13,14,19,24)(15,26,29,34)(16,31,32,37)(17,23,18,22)(20,25,36,28)(21,30,38,33)(27,42,41,35)(39,44,43,40);(13,15,17,25,19,29,18,28)(14,20,22,34,24,36,23,26)(16,27,30,39,32,41,33,43)(21,35,37,44,38,42,31,40);(1,2,3,4)(13,16,18,33,19,32,17,30)(14,21,23,31,24,38,22,37)(15,27,28,43,29,41,25,39)(20,35,26,40,36,42,34,44);(5,8,9,7)(6,11,12,10)(16,32)(21,38)(27,41)(30,33)(31,37)(35,42)(39,43)(40,44)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521382);