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 | |
4173 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)(3,9)(4,8)(5,7)(6,13)(10,14)(11,15)(12,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,40,43)(35,44,38,41)(36,39,37,42)(45,46,47,48);(2,5)(8,9)(10,13)(15,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,40,38)(34,41,43,44)(36,47,37,45)(39,46,42,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,36,40,37)(34,42,43,39)(35,45,38,47)(41,46,44,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,37,40,36)(34,39,43,42)(35,45,38,47)(41,46,44,48);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,38,40,35)(34,41,43,44)(36,47,37,45)(39,48,42,46);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,39)(34,36)(35,46)(37,43)(38,48)(40,42)(41,47)(44,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327580);