Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34636 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,24)(19,20,30,25)(21,33,34,26)(22,36,37,27)(28,38,41,31)(29,39,43,32)(35,40,46,45)(42,44,48,47);(2,3)(6,7)(9,10,12,14)(11,13,16,15)(18,24)(20,25)(21,34)(27,36)(28,41)(32,39)(35,46)(42,48);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,30)(18,25,24,20)(21,28,34,41)(22,29,37,43)(26,38,33,31)(27,39,36,32)(35,42,46,48)(40,47,45,44);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,23,25)(18,19,24,30)(21,31,34,38)(22,32,37,39)(26,28,33,41)(27,29,36,43)(35,44,46,47)(40,42,45,48);(1,4,5,8)(2,6,3,7)(17,21,23,34)(18,26,24,33)(19,28,30,41)(20,31,25,38)(22,35,37,46)(27,40,36,45)(29,42,43,48)(32,44,39,47);(17,22,23,37)(18,27,24,36)(19,29,30,43)(20,32,25,39)(21,35,34,46)(26,40,33,45)(28,42,41,48)(31,44,38,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022718);