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 | |
4235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,12,9)(5,10,13,8)(11,15,16,14)(17,18,22,24)(19,21,27,23)(20,28,29,25)(26,32,30,31)(33,34,38,43)(35,41,45,39)(36,44,46,40)(37,42,48,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,27)(18,23,24,21)(20,26,29,30)(25,32,28,31)(33,35)(34,39)(36,42)(37,44)(38,45)(40,48)(41,43)(46,47);(2,3)(5,13)(7,9)(11,16)(18,24)(20,29)(21,23)(26,30)(36,46)(37,48)(40,44)(42,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,22,29)(18,25,24,28)(19,26,27,30)(21,31,23,32)(33,36)(34,40)(35,42)(37,41)(38,46)(39,48)(43,44)(45,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,21)(20,30)(22,27)(23,24)(25,32)(26,29)(28,31)(33,35)(34,41)(36,47)(37,44)(38,45)(39,43)(40,48)(42,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,19)(20,31)(22,23)(24,27)(25,26)(28,30)(29,32)(33,37,38,48)(34,42,43,47)(35,44,45,40)(36,41,46,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8814181);