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 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,16)(4,12,14,5)(10,11,13,15)(17,18,21,23)(19,25,27,22)(20,28,30,24)(26,31,29,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3)(2,7)(4,10)(5,15)(6,9)(8,16)(11,12)(13,14)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(3,9)(7,16)(10,13)(11,15)(18,23)(22,25)(24,28)(31,32)(33,34,38,41)(35,36,39,42)(37,40,46,48)(43,44,45,47);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,27)(26,32)(29,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,5)(3,10)(6,14)(7,15)(8,12)(9,13)(11,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5)(2,4)(3,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8733428);