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 | |
4093 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)(3,5)(4,8)(6,7)(9,14)(10,13)(11,15)(12,16)(17,18,22,25)(19,23,27,21)(20,24,28,31)(26,29,32,30)(35,41)(39,42)(43,47)(46,48);(2,7)(3,9)(4,11)(12,13)(17,19,22,27)(18,21,25,23)(20,26,28,32)(24,30,31,29)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(2,7)(3,9)(4,11)(12,13)(20,28)(24,31)(26,32)(29,30)(34,38)(39,42)(40,44)(46,48);(1,3,6,9)(2,5,7,14)(4,10,11,16)(8,13,15,12)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,32)(30,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,4,6,11)(2,8,7,15)(3,10,9,16)(5,13,14,12)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,30,23,29)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(1,5)(2,9)(3,7)(4,12)(6,14)(8,10)(11,13)(15,16)(17,21,22,23)(18,19,25,27)(20,29,28,30)(24,32,31,26)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=g,f^2=gh,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, 9930329);