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 | |
4276 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)(3,5)(4,7)(6,12)(8,9)(10,14)(11,15)(13,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34)(35,39)(36,43)(37,45)(38,41)(40,46)(42,44)(47,48);(1,3,7,9)(2,8,4,5)(6,11,14,16)(10,15,12,13)(18,19)(23,25)(24,26)(30,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,4)(5,8)(10,12)(13,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34,38,41)(35,36,43,39)(37,40,46,45)(42,44,48,47);(1,4,7,2)(3,8,9,5)(6,12,14,10)(11,15,16,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,5,7,8)(2,9,4,3)(6,13,14,15)(10,16,12,11)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,6,7,14)(2,10,4,12)(3,11,9,16)(5,13,8,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034755);