Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
591 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)(5,8)(7,14)(10,15,13,16)(18,23,26,25)(24,31,32,27);(1,3)(2,6)(4,13,14,10)(5,11)(7,16,12,15)(8,9)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,32)(30,31);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(17,19,20,28,22,30,21,29)(18,24,23,31,26,32,25,27)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fhi,g^b=ghi,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28014);