Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#3194 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1106 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)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18,22,26)(19,27)(20,23,21,25)(24,30)(28,32)(29,31);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,14)(13,16)(18,23)(19,28)(20,21)(25,26)(29,30)(31,32);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21610);