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 | |
4039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,12,13,8)(11,14,16,15)(17,18,22,27)(19,25,28,23)(20,24,26,21)(29,30,31,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(2,4)(7,9)(8,12)(14,15)(19,28)(23,25)(29,31)(30,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,4)(3,10)(5,13)(14,15)(17,19,22,28)(18,23,27,25)(20,31,26,29)(21,32,24,30)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,20,22,26)(18,24,27,21)(19,29,28,31)(23,32,25,30)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,19)(18,25)(20,31)(21,30)(22,28)(23,27)(24,32)(26,29)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,26,27,20)(19,30,28,32)(23,31,25,29)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=gi,e^2=g,f^2=h,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, 8900217);