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 | |
3962 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(19,26)(24,27)(28,31)(29,32)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,18,22,23)(19,27,26,24)(20,21,25,30)(28,32,31,29)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,7)(4,10)(9,13)(12,14)(18,23)(20,25)(24,27)(28,31)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,19,22,26)(18,24,23,27)(20,28,25,31)(21,29,30,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,20)(18,21)(19,28)(22,25)(23,30)(24,29)(26,31)(27,32)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,32)(27,28)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9949790);