Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519056 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58997 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,15)(4,20)(5,23)(6,27)(7,13)(8,12)(9,14)(10,18)(11,25)(16,28)(17,21)(19,32)(22,29)(24,31)(26,30);(1,3,7,18,9,19,8,29)(2,10,12,15,14,22,13,32)(4,21,23,16,24,11,30,27)(5,25,20,28,26,17,31,6);(1,4)(2,5)(3,16)(6,22)(7,23)(8,30)(9,24)(10,28)(11,18)(12,20)(13,31)(14,26)(15,17)(19,27)(21,29)(25,32);(1,5)(2,4)(3,17)(6,18)(7,20)(8,31)(9,26)(10,21)(11,22)(12,23)(13,30)(14,24)(15,16)(19,25)(27,32)(28,29);(1,6,7,25,9,28,8,17)(2,11,12,27,14,21,13,16)(3,5,18,20,19,26,29,31)(4,22,23,32,24,10,30,15)Polycyclic presentation:
2^9|b^2=f,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426252);