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 |
| prime radical | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4002 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(18,23)(19,25)(24,30)(26,31);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(2,9)(3,4)(5,15)(6,16)(19,25)(20,27)(21,29)(24,30)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901588);