Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #38035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1525 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)(17,18,23,35,22,27,19,24)(20,30,28,25,33,37,32,40)(21,26,36,46,34,41,29,38)(31,43,42,39,44,47,45,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,27,35)(20,28,33,32)(21,29,34,36)(25,37,40,30)(26,38,41,46)(31,42,44,45)(39,47,48,43);(1,2)(3,8)(4,6)(5,7)(11,16)(13,15)(17,20,22,33)(18,25,27,40)(19,28,23,32)(21,31,34,44)(24,37,35,30)(26,39,41,48)(29,42,36,45)(38,47,46,43);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,21)(18,26)(19,29)(20,31)(22,34)(23,36)(24,38)(25,39)(27,41)(28,42)(30,43)(32,45)(33,44)(35,46)(37,47)(40,48);(1,4,5,3)(2,6,7,8)(21,34)(26,41)(29,36)(31,44)(38,46)(39,48)(42,45)(43,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=i,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6896331);