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 | |
4010 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)(3,8)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19,22,21)(18,23,26,25)(20,28,30,27)(24,32,29,31)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(2,4)(3,9)(10,12)(11,16)(17,18,22,26)(19,23,21,25)(20,24,30,29)(27,31,28,32)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,18)(19,23)(20,29)(21,25)(22,26)(24,30)(27,32)(28,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,21,22,19)(18,25,26,23)(20,27,30,28)(24,31,29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113759);