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 | #49704 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3944 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18,22,23)(19,21,26,24)(20,25,30,32)(27,31,28,29)(33,34,39,36)(35,40,42,38)(37,41,47,44)(43,46,45,48);(2,3)(4,11)(8,10)(12,13)(18,23)(19,26)(25,32)(27,28)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,48,44,46);(2,3)(7,9)(8,10)(15,16)(34,36)(35,42)(41,44)(43,45);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,36,39,34)(35,40,42,38)(37,41,47,44)(43,48,45,46);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,37)(34,41)(35,43)(36,44)(38,48)(39,47)(40,46)(42,45);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,38,39,40)(34,42,36,35)(37,46,47,48)(41,43,44,45)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=gi,e^2=g,f^2=gh,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, 10154619);