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 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(2,4)(7,10)(8,12)(14,16)(20,27)(24,30)(26,31)(28,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(3,9)(7,10)(11,15)(14,16)(18,23)(19,25)(24,30)(26,31)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877345);