Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120868 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1442 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,5)(6,9)(7,10)(8,13)(12,14)(15,16)(17,18)(19,25)(20,28,30,23)(21,24)(22,27)(26,31,29,32);(1,3,6,14,7,15,5,13)(2,8,4,16,10,12,9,11)(17,19)(18,22,24,25)(20,29)(21,27)(23,32,28,31)(26,30);(1,4,5,2,7,9,6,10)(3,12,13,16,15,8,14,11)(17,20)(18,23)(19,26,27,29)(21,30)(22,31,25,32)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fh,f^c=fhi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28064);