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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,40,38)(35,41,37,46)(36,43,39,42)(44,48,45,47);(2,4)(3,9)(8,12)(11,15)(19,25)(22,29)(26,31)(30,32)(33,35)(34,41)(36,45)(37,40)(38,46)(39,44)(42,48)(43,47);(2,4)(3,9)(8,12)(11,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,40,39)(34,42,38,43)(35,44,37,45)(41,47,46,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,35)(34,41,38,46)(36,44,39,45)(42,48,43,47);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,38,40,34)(35,41,37,46)(36,42,39,43)(44,48,45,47);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,39,40,36)(34,43,38,42)(35,44,37,45)(41,47,46,48)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,d^2=i,e^2=h,f^2=h,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, 9947997);