Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,11)(12,16)(17,18,24,32,23,27,22,21)(19,28,29,25,26,31,30,20);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,23,26)(18,25,27,20)(21,31,32,28)(22,29,24,30);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20,23,25)(18,26,27,19)(21,29,32,30)(22,31,24,28);(3,8)(5,7)(10,11)(14,15)(17,21,24,18,23,32,22,27)(19,20,29,28,26,25,30,31);(9,12,13,16)(10,14,11,15)(17,22,23,24)(18,21,27,32)(19,29,26,30)(20,28,25,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=hi,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104730);