Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | #35210 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,22,26,23,27,21,35)(19,24,31,39,29,38,30,45)(20,32,33,41,34,40,36,25)(28,42,44,47,43,48,46,37);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,30,22,31)(25,37,41,47)(26,39,35,45)(32,42,40,48)(33,44,36,46);(11,16)(13,15)(17,20,23,34)(18,25,27,41)(19,28,29,43)(21,33,22,36)(24,37,38,47)(26,40,35,32)(30,44,31,46)(39,48,45,42);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(19,29)(24,38)(28,43)(30,31)(37,47)(39,45)(42,48)(44,46);(1,4,5,3)(2,6,7,8)(17,21,23,22)(18,26,27,35)(19,30,29,31)(20,33,34,36)(24,39,38,45)(25,40,41,32)(28,44,43,46)(37,48,47,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6781362);