Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 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)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,22)(21,31)(23,28)(24,27)(25,30)(29,32);(2,6)(7,8)(10,13)(11,16)(17,19)(18,25)(20,31)(21,24)(22,29)(23,30)(26,28)(27,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,18,23,27,24,28)(19,26,21,32,30,25,29,31);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,22,30,23,29,24,19)(18,26,27,32,28,25,20,31);(2,6)(7,8)(10,13)(14,15)(17,22,23,24)(18,27,28,20)(19,29,30,21)(25,32,26,31)Polycyclic presentation:
2^9|c^2=fh,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503178);