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 | |
4102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34)(35,38)(36,45)(37,41)(39,43)(40,46)(42,47)(44,48);(1,2,6,4)(3,10,9,7)(5,8,13,12)(11,16,15,14)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35,39,42)(34,38,43,47)(36,37,46,44)(40,48,45,41);(2,4)(7,10)(8,12)(14,16)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,39,46)(34,40,43,45)(35,44,42,37)(38,41,47,48);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36)(34,40)(35,44)(37,42)(38,41)(39,46)(43,45)(47,48);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,37,39,44)(34,41,43,48)(35,36,42,46)(38,45,47,40);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38)(34,42)(35,43)(36,41)(37,40)(39,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=i,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, 9976290);