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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,11,13,7)(4,8,15,12)(5,9,16,14)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28);(1,3,6,13)(2,7,10,11)(4,9,15,14)(5,12,16,8)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,15)(10,11)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4)(2,8)(3,9)(5,7)(6,15)(10,12)(11,16)(13,14)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,5)(2,9)(3,12)(4,11)(6,16)(7,15)(8,13)(10,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154191);