Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1355 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(24,29)(27,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,24)(19,26)(20,28)(22,31)(23,29)(25,30);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,18,21,23)(19,20,26,30)(22,29,31,24)(25,27,28,32);(1,3,5,4)(2,7,8,6)(17,19,21,26)(18,25,23,28)(20,29,30,24)(22,32,31,27);(1,4,5,3)(2,6,8,7)(17,20)(18,26)(19,23)(21,30)(22,28)(24,27)(25,31)(29,32)Polycyclic presentation:
2^9|b^2=fgi,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490218);