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) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,12,16)(4,6,11,14)(5,13,15,10)(17,18)(19,25)(20,30)(21,26)(22,27)(23,24)(28,29)(31,32)(33,34)(35,42)(36,44)(37,39)(38,40)(41,46)(43,47)(45,48);(1,3,7,12)(2,8,9,16)(4,10,11,13)(5,14,15,6)(18,24)(21,27)(25,30)(28,31)(33,35,39,38)(34,40,37,42)(36,45,46,43)(41,48,44,47);(2,9)(3,12)(6,14)(10,13)(17,19,23,20)(18,25,24,30)(21,28,27,31)(22,32,26,29)(33,34,39,37)(35,40,38,42)(36,41,46,44)(43,47,45,48);(1,4,7,11)(2,6,9,14)(3,13,12,10)(5,8,15,16)(17,20,23,19)(18,25,24,30)(21,31,27,28)(22,32,26,29)(33,36)(34,41)(35,43)(37,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,10)(3,6)(4,8)(7,15)(9,13)(11,16)(12,14)(17,21,23,27)(18,26,24,22)(19,28,20,31)(25,29,30,32)(33,37,39,34)(35,40,38,42)(36,44,46,41)(43,47,45,48);(1,6)(2,11)(3,5)(4,9)(7,14)(8,13)(10,16)(12,15)(17,22,23,26)(18,27,24,21)(19,29,20,32)(25,28,30,31)(33,38,39,35)(34,40,37,42)(36,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109421);