Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C2 | C2×#1464 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #117 ◁ #2873 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,5,10,17,25,6,7,16,20,18,21,31,32)(3,11,13,27,14,19,29,24,12,26,15,30,28,8,22,23);(2,7)(8,19)(10,21)(11,26);(1,3)(2,8,7,19)(4,15)(5,14)(6,12)(9,24,20,23)(10,11,21,26)(13,16)(17,22)(18,28)(25,30,32,27)(29,31);(3,12)(8,19)(11,26)(13,15)(14,28)(22,29)(23,24)(27,30)Polycyclic presentation:
2^9|a^2=f,c^2=g,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,f^b=fg,d^c=di,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292717, #72292724, C2×#480320 |
| maximal normal subgroup of | #72292717, #72292724, C2×#480320 |
| maximal quotient of | C2×#480320, #72292838 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387516);