Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24435 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1586 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)(19,23)(20,32)(21,25)(22,27)(24,35)(26,34)(28,44)(29,38)(30,39)(31,43)(33,47)(36,41)(37,46)(40,45)(42,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,27,38)(20,33,36,28)(21,31,34,30)(24,40,42,37)(25,43,26,39)(32,47,41,44)(35,45,48,46);(2,6)(5,7)(10,13)(14,16)(17,20,34,48,22,36,21,35)(18,24,25,41,27,42,26,32)(19,28,30,45,29,33,31,46)(23,37,43,47,38,40,39,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(23,38)(28,33)(30,31)(37,40)(39,43)(44,47)(45,46);(3,8)(5,7)(10,13)(14,16)(17,21,22,34)(18,25,27,26)(19,30,29,31)(20,35,36,48)(23,39,38,43)(24,41,42,32)(28,45,33,46)(37,44,40,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=hi,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=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6691855);