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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,26)(19,28,30,24)(20,33,35,41)(21,27,23,37)(25,34,40,36)(29,44,45,48)(31,43,32,39)(38,47,42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,29,35,45)(21,31,23,32)(25,38,40,42)(27,39,37,43)(33,48,41,44)(34,46,36,47);(3,8)(5,7)(10,13)(14,16)(17,20,22,35)(18,25,26,40)(19,29,30,45)(21,36,23,34)(24,38,28,42)(27,41,37,33)(31,47,32,46)(39,44,43,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(25,40)(27,37)(29,45)(31,32)(33,41)(46,47);(10,13)(14,16)(17,21,22,23)(18,27,26,37)(19,31,30,32)(20,34,35,36)(24,39,28,43)(25,33,40,41)(29,46,45,47)(38,48,42,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678735);