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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,5,4,8)(6,9,14,16)(11,13,12,15)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(1,3,7,4)(2,8,10,5)(6,11,14,12)(9,15,16,13)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(6,14)(9,16)(11,12)(13,15)(19,26)(22,24)(27,29)(30,32)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,4,7,3)(2,5,10,8)(6,12,14,11)(9,13,16,15)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,5)(2,4)(3,10)(6,13)(7,8)(9,12)(11,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,6,7,14)(2,9,10,16)(3,11,4,12)(5,13,8,15)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8267062);