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 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,8,13,12)(10,15,16,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(34,40)(35,37)(36,45)(44,48);(2,4)(7,9)(8,12)(14,15)(19,20)(21,30)(23,26)(24,29)(33,34)(35,41)(36,38)(37,47)(39,40)(42,45)(43,48)(44,46);(5,13)(8,12)(10,16)(14,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,39,37)(34,41,40,47)(36,43,45,46)(38,48,42,44);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36)(34,42)(35,43)(37,46)(38,40)(39,45)(41,44)(47,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37,39,35)(34,41,40,47)(36,46,45,43)(38,48,42,44);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,38,39,42)(34,36,40,45)(35,44,37,48)(41,46,47,43)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=gh,e^2=g,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=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109610);