Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #507264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28984 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13310 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,11)(6,12)(7,24)(8,14)(9,15)(10,19)(13,27)(16,32)(17,21)(18,22)(20,25)(23,29)(26,28)(30,31);(1,3,7,25,6,10,23,31,8,17,27,16,5,15,22,28)(2,9,13,30,12,4,18,32,14,19,24,26,11,21,29,20);(1,4)(2,10)(3,12)(5,19)(6,9)(7,26)(8,21)(11,17)(13,16)(14,15)(18,31)(20,22)(23,32)(24,25)(27,30)(28,29);(2,11,14,12)(3,10,17,15)(7,27)(13,18,24,29)(16,31,25,28)(20,32)(22,23)(26,30)Polycyclic presentation:
2^9|b^2=efg,d^2=fh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gh,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414460);