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 | |
4189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,34,39,38)(35,40,37,42)(36,41,46,44)(43,47,45,48);(2,9)(3,4)(10,16)(11,12)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31)(33,37,39,35)(34,40,38,42)(36,43,46,45)(41,48,44,47);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,29)(30,32)(33,38,39,34)(35,42,37,40)(36,44,46,41)(43,48,45,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10102638);