Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1755 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)(3,8)(4,6)(5,7)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(9,11)(10,13)(12,16)(14,15)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(11,16)(13,15)(18,23)(19,26)(24,31)(25,28);(1,4,5,3)(2,6,7,8)(10,14)(13,15)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6409523);