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 | |
4163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,4,10)(5,13,15,11)(6,14,16,12)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,28)(29,32)(37,45)(40,46)(42,47)(43,48);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,46)(47,48);(2,9)(3,4)(11,13)(12,14)(18,24)(20,25)(26,30)(28,32)(34,39)(35,41)(40,46)(42,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=i,e^2=h,f^2=g,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066772);