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 | |
4105 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34)(35,43)(36,46)(37,48)(38,39)(40,45)(41,47)(42,44);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(19,23)(20,27)(26,31)(28,32)(33,34,39,38)(35,40,45,43)(36,41,47,46)(37,48,44,42);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,14,16,11)(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,35,39,45)(34,40,38,43)(36,44,47,37)(41,42,46,48);(1,3)(2,4)(5,10)(6,14)(7,8)(9,13)(11,15)(12,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,36)(34,41)(35,44)(37,45)(38,46)(39,47)(40,42)(43,48);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,14,16,11)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,37,39,44)(34,42,38,48)(35,36,45,47)(40,46,43,41);(1,6,7,15)(2,9,10,16)(3,11,8,14)(4,12,5,13)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,38,39,34)(35,43,45,40)(36,41,47,46)(37,48,44,42)Polycyclic presentation:
2^9|b^2=h,c^2=gi,e^2=g,f^2=g,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, 10330217);