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 | |
4353 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)(3,8)(4,10)(5,14)(6,11)(7,9)(12,16)(13,15)(18,23)(19,25)(24,30)(26,31)(34,39)(35,36)(41,46)(43,44);(1,3,7,5)(2,8,9,14)(4,12,13,11)(6,10,16,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(2,9)(3,5)(10,15)(11,12)(20,27)(24,30)(26,31)(28,32)(33,35,38,36)(34,40,39,42)(37,43,45,44)(41,47,46,48);(1,4,7,13)(2,10,9,15)(3,11,5,12)(6,14,16,8)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,35)(34,40,39,42)(37,43,45,44)(41,48,46,47);(1,5,7,3)(2,8,9,14)(4,12,13,11)(6,15,16,10)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(1,6)(2,11)(3,10)(4,8)(5,15)(7,16)(9,12)(13,14)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441751);