Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14,19,25)(15,23,28,21)(16,30,34,42)(17,36,29,38)(18,26,20,24)(22,35,41,33)(27,43,44,39)(31,40,32,37);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,31)(17,18)(19,28)(20,29)(22,39)(23,25)(24,36)(26,38)(27,33)(30,37)(32,34)(35,44)(40,42)(41,43);(13,16,20,35,19,34,18,33)(14,22,26,42,25,41,24,30)(15,27,17,32,28,44,29,31)(21,37,36,43,23,40,38,39);(2,4)(13,17,18,15,19,29,20,28)(14,23,24,36,25,21,26,38)(16,32,33,27,34,31,35,44)(22,40,30,43,41,37,42,39);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,24,25,26)(15,17,28,29)(16,33,34,35)(21,36,23,38)(22,30,41,42)(27,32,44,31)(37,43,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=g,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=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470722);