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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3851 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)(3,11)(4,6)(5,8)(7,10)(9,15)(12,16)(13,14)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,39,37)(35,42,41,38)(36,44,47,40)(43,46,45,48);(1,3,7,5)(2,8,10,11)(4,12,15,13)(6,14,9,16)(18,24)(19,21)(26,30)(27,29)(33,35,39,41)(34,38,37,42)(36,45,47,43)(40,48,44,46);(1,4)(2,9)(3,12)(5,13)(6,10)(7,15)(8,16)(11,14)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,34,39,37)(35,38,41,42)(36,44,47,40)(43,48,45,46);(1,4)(2,9)(3,13)(5,12)(6,10)(7,15)(8,14)(11,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,36,39,47)(34,40,37,44)(35,43,41,45)(38,46,42,48);(1,5,7,3)(2,8,10,11)(4,13,15,12)(6,14,9,16)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37,39,34)(35,38,41,42)(36,40,47,44)(43,48,45,46);(1,6,7,9)(2,4,10,15)(3,14,5,16)(8,12,11,13)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,38)(34,41)(35,37)(36,46)(39,42)(40,45)(43,44)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227595);