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 | |
3996 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)(3,8,5,6)(4,9,13,16)(11,15,14,12)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(3,5)(6,8)(11,14)(12,15)(18,25)(19,30)(20,27)(28,32)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9452757);