Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1638 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)(17,18)(19,29)(20,32)(21,27)(22,24)(23,28)(25,30)(26,31);(9,10,12,13)(11,14,15,16)(18,24)(19,30)(21,22)(25,26)(27,28)(29,32);(1,3,5,4)(2,6,8,7)(10,13)(11,15)(17,19)(18,25)(20,21)(22,31)(23,30)(24,32)(26,27)(28,29);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,23,31)(18,26,28,32)(19,21,30,22)(24,25,27,29);(1,4,5,3)(2,6,8,7)(10,13)(11,15)(17,21,23,22)(18,27,28,24)(19,31,30,20)(25,32,29,26)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6523487);