Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #23853 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,28)(21,27)(22,25)(24,31)(26,32)(29,30);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,27)(18,21,25,23)(20,24,31,28)(26,30,29,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19)(18,21)(20,29)(22,27)(23,25)(24,30)(26,31)(28,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,22,31)(18,24,25,28)(19,26,27,29)(21,32,23,30);(2,6)(7,8)(17,21,22,23)(18,19,25,27)(20,30,31,32)(24,29,28,26)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433232);