Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,12)(11,14,16,15)(18,23)(19,27)(20,21)(24,26)(25,32)(28,29);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(18,24)(19,28)(20,21)(23,26)(25,32)(27,29);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,18,20,23,22,26,21,24)(19,25,27,30,29,32,28,31);(1,3,5,4)(2,7,8,6)(11,16)(14,15)(17,19,20,27,22,29,21,28)(18,25,23,30,26,32,24,31);(1,4,5,3)(2,7,8,6)(19,29)(25,32)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fhi,d^2=fg,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, 7512940);