Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #45063 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,24,28,22,31)(19,25,20,26,29,32,21,30);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(17,19,24,29)(18,25,28,32)(20,22,21,23)(26,31,30,27);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,23)(21,24)(22,29)(26,31)(27,30)(28,32);(2,6)(7,8)(11,16)(13,15)(17,21,24,20)(18,26,28,30)(19,22,29,23)(25,27,32,31);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,22,24,23)(18,27,28,31)(19,20,29,21)(25,30,32,26)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6921022);