Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122122 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
723 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)(3,10,12,6)(4,9)(5,8)(7,11,15,13)(14,16)(17,18)(19,22,26,30)(20,27)(21,24)(23,28)(25,31,32,29);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,22,24,30)(20,25)(21,26)(23,31,27,29)(28,32);(1,3)(2,7)(4,13)(5,12)(6,9)(8,15)(10,16)(11,14)(17,20)(18,23,24,27)(19,25)(21,28)(22,29,30,31)(26,32)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,f^a=fi,c^b=cf,d^b=di,g^b=gh,d^c=di,e^c=eh,g^c=ghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29318);