Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519061 |
| 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 | |
2484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,9,15)(3,16,10,29)(4,19,22,18)(5,23,25,21)(6,26,28,32)(7,14,8,13)(11,30,31,24)(12,17,20,27);(1,3,7,18,9,10,8,19)(2,4,13,16,15,22,14,29)(5,17,24,6,25,27,30,28)(11,12,23,26,31,20,21,32);(1,4)(2,10)(3,15)(5,20)(6,21)(7,16)(8,29)(9,22)(11,27)(12,25)(13,19)(14,18)(17,31)(23,28)(24,32)(26,30);(1,5)(2,11)(3,17)(4,20)(6,18)(7,24)(8,30)(9,25)(10,27)(12,22)(13,23)(14,21)(15,31)(16,32)(19,28)(26,29);(1,6,7,27,9,28,8,17)(2,12,13,26,15,20,14,32)(3,5,18,24,10,25,19,30)(4,21,16,11,22,23,29,31)Polycyclic presentation:
2^9|a^2=i,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,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426258);