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 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,9,14,15)(11,12,13,16)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(2,7)(3,10)(9,15)(11,13)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(2,7)(4,8)(9,15)(12,16)(20,28)(24,31)(27,29)(30,32)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,14)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,4,6,8)(2,3,7,10)(5,12,14,16)(9,11,15,13)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=i,e^2=h,f^2=gh,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, 8440751);