Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907945 |
| 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 | |
1507 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)(19,24)(20,32)(21,26)(22,36)(23,27)(25,33)(28,42)(29,38)(30,39)(31,46)(34,40)(35,41)(37,44)(43,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,30,22,31)(25,37,41,47)(26,39,36,46)(32,42,40,48)(33,44,35,45);(3,8)(5,7)(10,13)(11,15)(17,20,22,33,23,34,21,35)(18,25,26,32,27,41,36,40)(19,28,31,44,29,43,30,45)(24,37,39,42,38,47,46,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,43)(30,31)(37,47)(39,46)(42,48)(44,45);(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,34,35)(24,39,38,46)(25,40,41,32)(28,44,43,45)(37,48,47,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fh,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=efh,d^c=dfh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6847390);