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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(17,18,23,25)(19,24,21,22)(20,26,30,32)(27,29,28,31)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(2,6)(3,10)(8,14)(11,12)(17,19)(18,24)(20,28)(21,23)(22,25)(26,31)(27,30)(29,32)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,6)(7,9)(8,14)(15,16)(18,25)(19,21)(26,32)(27,28)(35,40)(37,45)(43,46)(44,48);(1,3)(2,7)(4,12)(5,10)(6,9)(8,16)(11,13)(14,15)(17,20)(18,26)(19,27)(21,28)(22,31)(23,30)(24,29)(25,32)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21,23,19)(18,24,25,22)(20,28,30,27)(26,29,32,31)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,22,23,24)(18,19,25,21)(20,29,30,31)(26,28,32,27)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529172);