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 | |
4110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,11)(5,8,14,16)(10,12,13,15)(17,18,21,23)(19,25,27,22)(20,28,30,24)(26,31,29,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,48)(43,45)(44,47);(1,3)(2,4)(5,13)(6,7)(8,15)(9,11)(10,14)(12,16)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,39,40)(34,36,42,37)(38,47,48,43)(41,45,46,44);(3,7)(4,11)(10,13)(12,15)(18,23)(22,25)(24,28)(31,32)(33,36,39,37)(34,40,42,35)(38,44,48,45)(41,47,46,43);(1,4,6,11)(2,7,9,3)(5,12,14,15)(8,13,16,10)(17,18)(19,22)(20,28)(21,23)(24,30)(25,27)(26,32)(29,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,37,39,36)(34,35,42,40)(38,45,48,44)(41,43,46,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,16)(11,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,38,39,48)(34,41,42,46)(35,43,40,47)(36,44,37,45)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748611);