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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,21,24)(19,22,26,31)(20,27,29,23)(25,32,28,30)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(3,9)(6,12)(10,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(28,32)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,31)(27,32)(28,29)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(28,32)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034767);