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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,26)(20,25)(21,27)(22,32)(23,24)(28,31)(29,30)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(18,24)(21,32)(25,26)(28,30)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(2,6)(7,13)(8,14)(15,16)(17,19,23,20)(18,25,24,26)(21,28,32,30)(22,31,27,29)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20,23,19)(18,26,24,25)(21,28,32,30)(22,31,27,29);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21)(18,27)(19,28)(20,30)(22,24)(23,32)(25,29)(26,31)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,22,23,27)(18,21,24,32)(19,29,20,31)(25,30,26,28)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808835);