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 | |
4083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31)(33,34)(35,40)(36,39)(37,41)(38,44)(42,47)(43,45)(46,48);(2,7)(4,8)(10,13)(12,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(33,35,39,41)(34,40,36,37)(38,43,47,48)(42,46,44,45);(2,7)(3,9)(10,13)(11,16)(20,28)(24,32)(27,29)(30,31)(34,36)(35,41)(42,44)(43,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,13)(15,16)(17,20,22,28)(18,24,26,32)(19,27,25,29)(21,30,23,31)(33,36,39,34)(35,37,41,40)(38,44,47,42)(43,45,48,46);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,19)(18,21)(20,29)(22,25)(23,26)(24,31)(27,28)(30,32)(33,37)(34,41)(35,36)(38,46)(39,40)(42,43)(44,48)(45,47);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,21)(18,25)(19,26)(20,30)(22,23)(24,29)(27,32)(28,31)(33,38,39,47)(34,42,36,44)(35,43,41,48)(37,45,40,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=gh,e^2=hi,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, 9113745);