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 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(19,25)(23,26)(27,31)(30,32)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35,38,44)(34,39,42,48)(36,37,46,43)(40,47,45,41);(2,7)(3,10)(8,15)(11,12)(18,20)(23,26)(24,28)(30,32)(33,36,38,46)(34,40,42,45)(35,43,44,37)(39,41,48,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,40)(35,43)(37,44)(38,46)(39,41)(42,45)(47,48);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,38,43)(34,41,42,47)(35,36,44,46)(39,45,48,40);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976323);