Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,8,13,15,11,16,12,14)(17,18)(19,25,27,22)(20,23)(21,24)(26,30,28,31)(29,32);(2,7,10,9)(3,11)(8,15,16,14)(12,13)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(1,3,4,12,6,11,5,13)(2,8,7,14,10,16,9,15)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18325);