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 | |
4203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,12,13,8)(11,14,16,15)(17,18,22,24)(19,23,21,26)(20,25,28,31)(27,30,29,32)(34,36)(39,43)(40,44)(47,48);(2,4)(7,9)(8,12)(14,15)(17,19)(18,23)(20,27)(21,22)(24,26)(25,30)(28,29)(31,32)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(2,4)(3,10)(5,13)(14,15)(18,24)(19,21)(20,28)(30,32)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,20,22,28)(18,25,24,31)(19,27,21,29)(23,30,26,32)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,21,22,19)(18,26,24,23)(20,27,28,29)(25,30,31,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901786);