Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18,22,27)(19,30,32,25)(20,34,37,46)(21,40,33,42)(23,29,24,28)(26,39,45,38)(31,47,48,43)(35,44,36,41);(2,6)(7,8)(10,13)(11,16)(17,19,22,32)(18,25,27,30)(20,35,37,36)(21,23,33,24)(26,43,45,47)(28,42,29,40)(31,39,48,38)(34,41,46,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,24,39,22,37,23,38)(18,26,29,46,27,45,28,34)(19,31,33,36,32,48,21,35)(25,41,42,47,30,44,40,43);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,24,19,22,33,23,32)(18,25,29,42,27,30,28,40)(20,36,39,48,37,35,38,31)(26,44,46,43,45,41,34,47);(2,6)(7,8)(10,13)(14,15)(18,27)(20,37)(25,30)(28,29)(31,48)(35,36)(38,39)(40,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fhi,d^2=fg,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^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510673);