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 | |
4042 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)(17,18,22,25)(19,21,27,23)(20,24,30,32)(26,31,28,29)(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)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,31,32,29)(33,35,39,42)(34,38,43,47)(36,37,46,44)(40,48,45,41);(2,4)(7,10)(8,12)(14,16)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,31,32,29)(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)(18,21)(20,26)(22,27)(23,25)(24,31)(28,30)(29,32)(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)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28)(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)(17,21)(18,19)(20,29)(22,23)(24,28)(25,27)(26,32)(30,31)(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=ghi,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, 9976298);