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 | |
4112 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)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18)(19,24)(20,28)(21,22)(23,27)(25,29)(26,30)(31,32)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(18,22)(23,24)(25,28)(30,31)(33,35)(34,36)(37,39)(38,43)(40,42)(41,44)(45,47)(46,48);(3,11)(7,8)(12,14)(13,16)(17,19,21,27)(18,23,22,24)(20,26,29,32)(25,30,28,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,19)(18,24)(20,26)(21,27)(22,23)(25,31)(28,30)(29,32)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,46)(47,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,31)(30,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,46,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,20,21,29)(18,25,22,28)(19,26,27,32)(23,30,24,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=gh,f^2=g,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, 9078812);