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 | |
4172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,9,13,16)(11,12,15,14)(17,18,23,26)(19,24,27,22)(20,21,25,32)(28,29,30,31)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3,6,8)(2,7,10,4)(5,11,13,15)(9,14,16,12)(17,19,23,27)(18,24,26,22)(20,30,25,28)(21,31,32,29)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(5,13)(9,16)(11,15)(12,14)(19,27)(22,24)(28,30)(29,31)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,2)(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,20)(18,21)(19,28)(22,31)(23,25)(24,29)(26,32)(27,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,7)(9,15)(11,16)(13,14)(17,21)(18,25)(19,29)(20,26)(22,28)(23,32)(24,30)(27,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5,6,13)(2,9,10,16)(3,11,8,15)(4,12,7,14)(17,22)(18,19)(20,31)(21,28)(23,24)(25,29)(26,27)(30,32)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8296470);