Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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)(19,27)(20,32)(21,23)(22,36)(24,31)(25,33)(26,28)(29,45)(30,41)(34,42)(35,40)(37,43)(38,48)(39,46)(44,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(18,23)(20,33)(24,37)(26,36)(27,41)(29,34)(35,44)(46,48);(3,8)(5,7)(10,13)(11,15)(17,19,28,43,21,30,22,31)(18,24,36,41,23,37,26,27)(20,34,44,46,33,29,35,48)(25,39,47,42,32,38,40,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,33)(18,25,23,32)(19,29,30,34)(22,35,28,44)(24,38,37,39)(26,40,36,47)(27,42,41,45)(31,46,43,48);(10,13)(14,16)(19,30)(20,33)(24,37)(25,32)(27,41)(31,43)(35,44)(40,47)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6847561);