Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121865 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1445 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,10)(4,5)(8,11)(9,14)(15,16)(17,18)(19,22,26,30)(20,27)(21,24)(23,28)(25,29,31,32);(1,2,4,7,6,9,5,14)(3,11,13,8,10,15,12,16)(17,19)(18,22,24,30)(20,25)(21,26)(23,29,27,32)(28,31);(1,3,4,13,6,10,5,12)(2,8,7,15,9,16,14,11)(17,20)(18,23,24,27)(19,25)(21,28)(22,29,30,32)(26,31)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=eh,f^c=fhi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29061);