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 | |
4054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,14,8)(4,15,11,6)(5,10,16,13)(17,18)(19,24)(20,25)(21,28)(22,23)(26,32)(27,30)(29,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,7,14)(2,8,9,12)(4,13,11,10)(5,15,16,6)(17,18)(19,24)(20,29)(21,26)(22,23)(25,31)(27,30)(28,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,9)(4,11)(8,12)(10,13)(18,23)(19,27)(25,29)(26,28)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4,7,11)(2,6,9,15)(3,13,14,10)(5,12,16,8)(17,19,22,27)(18,24,23,30)(20,26,31,28)(21,25,32,29)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,5,7,16)(2,10,9,13)(3,6,14,15)(4,8,11,12)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,27)(30,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,6)(2,11)(3,5)(4,9)(7,15)(8,10)(12,13)(14,16)(17,21)(18,26)(19,25)(20,30)(22,32)(23,28)(24,31)(27,29)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10102678);