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 | |
4168 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,7)(6,13)(8,11)(14,15)(18,23)(19,26)(20,28)(30,31)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,21,28)(18,24,23,29)(19,27,26,32)(22,30,25,31)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,18)(19,25)(20,29)(21,23)(22,26)(24,28)(27,30)(31,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,19)(18,25)(20,27)(21,26)(22,23)(24,31)(28,32)(29,30)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,e^2=g,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, 9414037);