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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,12,5)(4,8,14,16)(10,15,11,13)(17,18,23,25)(19,21,27,20)(22,24,31,30)(26,29,32,28)(34,39)(36,40)(37,45)(43,46);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,19,23,27)(18,20,25,21)(22,26,31,32)(24,28,30,29)(34,39)(36,40)(42,47)(44,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,46)(45,47);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,18)(19,20)(21,27)(22,30)(23,25)(24,31)(26,29)(28,32)(33,35,38,41)(34,40,39,36)(37,46,45,43)(42,44,47,48);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,36,38,40)(34,41,39,35)(37,44,45,48)(42,46,47,43);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,22,23,31)(18,24,25,30)(19,26,27,32)(20,28,21,29)(33,37,38,45)(34,42,39,47)(35,43,41,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329856);