Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,30,21,27,22,26)(19,25,31,24,29,23,28,32);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,26)(25,27)(30,32);(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,24,27,32)(20,28,22,31)(23,30,25,26);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,19)(18,25)(20,31)(21,29)(22,28)(23,27)(24,30)(26,32);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,20,21,22)(18,26,27,30)(19,28,29,31)(23,32,25,24)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6923329);