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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,48,44,41);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,45)(46,47);(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35,38,45)(34,39,42,43)(36,44,47,37)(40,41,46,48);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36)(34,40)(35,44)(37,45)(38,47)(39,41)(42,46)(43,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,37,38,44)(34,41,42,48)(35,36,45,47)(39,46,43,40);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=h,e^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, 10326903);