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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3901 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,23,25)(19,20,27,21)(22,30,31,24)(26,29,32,28)(34,39)(35,42)(36,44)(45,47);(1,3,6,5)(2,7,8,10)(4,11,14,13)(9,15,12,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(2,8)(7,10)(9,12)(15,16)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,30,32)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,4)(2,9)(3,11)(5,13)(6,14)(7,15)(8,12)(10,16)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,6,3)(2,10,8,7)(4,13,14,11)(9,16,12,15)(17,22,23,31)(18,24,25,30)(19,26,27,32)(20,28,21,29)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103551);