Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1446 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
765 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)(3,6,10,13)(4,7,11,15)(5,8)(9,14)(12,16)(17,18)(19,26)(20,29)(21,25)(22,27)(23,28)(24,30)(31,32);(1,3)(2,6,8,13)(4,9)(5,10)(7,14,15,12)(11,16)(17,19)(18,22,25,26)(20,28)(21,27)(23,29,31,24)(30,32);(1,4)(2,7,8,15)(3,9)(5,11)(6,12,13,14)(10,16)(17,19)(18,23,25,31)(20,28)(21,27)(22,24,26,29)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29367);