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 |
| prime radical | #50830 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,12)(10,15)(13,16)(18,24)(20,25)(26,32)(28,30)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,23,24)(19,25,21,20)(22,26,31,32)(27,30,29,28)(33,35,38,42)(34,36,40,44)(37,41,47,45)(39,43,46,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,30,32,28)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,20)(18,19)(21,24)(22,30)(23,25)(26,29)(27,32)(28,31)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,21,23,19)(18,20,24,25)(22,27,31,29)(26,30,32,28)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,22,23,31)(18,26,24,32)(19,27,21,29)(20,28,25,30)(33,37,38,47)(34,39,40,46)(35,41,42,45)(36,43,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8570015);