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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(4,11)(5,13)(14,15)(17,18,23,27)(19,26,30,24)(20,22,21,25)(28,29,31,32)(33,34,38,41)(35,39,36,43)(37,40,46,45)(42,47,44,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,20,23,21)(18,25,27,22)(19,28,30,31)(24,29,26,32)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,21,23,20)(18,22,27,25)(19,28,30,31)(24,29,26,32)(33,34)(35,39)(36,43)(37,45)(38,41)(40,46)(42,48)(44,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,20)(19,29)(21,27)(23,25)(24,31)(26,28)(30,32)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 10329854);