Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,24,22,25,29,20)(19,26,27,31,28,32,30,23);(2,6)(7,8)(17,19,22,28)(18,23,25,31)(20,26,24,32)(21,30,29,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,18,22,24,29,25)(19,23,27,26,28,31,30,32);(1,4,5,3)(2,8,6,7)(17,20,29,25,22,24,21,18)(19,23,30,32,28,31,27,26);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=fi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366443);