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 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,4)(6,10,15,16)(11,14,13,12)(18,23)(19,25)(24,30)(26,31)(33,34,39,42)(35,43,45,38)(36,41,37,40)(44,47,46,48);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,14,16,12)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35)(34,38)(36,44)(37,46)(39,45)(40,47)(41,48)(42,43);(2,9)(3,5)(10,16)(11,13)(20,27)(24,30)(26,31)(28,32)(33,36,39,37)(34,40,42,41)(35,44,45,46)(38,47,43,48);(1,4,7,8)(2,3,9,5)(6,14,15,12)(10,13,16,11)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,37,39,36)(34,41,42,40)(35,44,45,46)(38,47,43,48);(1,5,7,3)(2,8,9,4)(6,13,15,11)(10,14,16,12)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,35)(34,38)(36,46)(37,44)(39,45)(40,48)(41,47)(42,43);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38)(34,35)(36,47)(37,48)(39,43)(40,44)(41,46)(42,45)Polycyclic presentation:
2^9|a^2=h,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, 8441815);