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 | |
4086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,30)(31,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,2,6,7)(3,4,10,9)(5,8,15,13)(11,12,16,14)(17,19,23,22)(18,24,27,28)(20,29,26,31)(21,30,25,32)(33,34,38,41)(35,36,39,42)(37,40,46,45)(43,44,47,48);(2,7)(3,10)(8,13)(11,16)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,32,28,30)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,32)(28,30)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,31,28,29)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,30,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113738);