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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3909 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)(3,7)(4,13)(5,10)(6,11)(8,12)(9,15)(14,16)(17,18,22,24)(19,23,27,28)(20,31,30,25)(21,32,29,26)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,3,6,12)(2,7,11,8)(4,14,15,10)(5,13,16,9)(17,19)(18,23)(20,29)(21,30)(22,27)(24,28)(25,32)(26,31)(34,36)(40,41)(42,44)(47,48);(3,12)(5,16)(7,8)(10,14)(18,24)(19,27)(21,29)(25,31)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(1,3)(2,8)(4,14)(5,13)(6,12)(7,11)(9,16)(10,15)(17,18)(19,28)(20,31)(21,26)(22,24)(23,27)(25,30)(29,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4,6,15)(2,9,11,13)(3,10,12,14)(5,7,16,8)(17,20,22,30)(18,25,24,31)(19,29,27,21)(23,32,28,26)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5)(2,10)(3,13)(4,7)(6,16)(8,15)(9,12)(11,14)(17,21,22,29)(18,26,24,32)(19,30,27,20)(23,31,28,25)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842074);