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 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,6)(4,5,10,9)(13,15,16,14)(17,18)(19,24)(20,30)(21,25)(22,31)(23,26)(27,28)(29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,3,7,12)(2,8,11,6)(4,13,10,16)(5,15,9,14)(17,19,23,29)(18,24,26,32)(20,25,31,27)(21,22,28,30)(34,39)(36,40)(41,45)(44,47);(3,12)(4,10)(5,9)(6,8)(17,20,23,31)(18,22,26,30)(19,27,29,25)(21,32,28,24)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,4)(2,9)(3,13)(5,11)(6,15)(7,10)(8,14)(12,16)(17,20)(18,22)(19,25)(21,32)(23,31)(24,28)(26,30)(27,29)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,5,7,9)(2,10,11,4)(3,14,12,15)(6,16,8,13)(17,21,23,28)(18,25,26,27)(19,22,29,30)(20,24,31,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,6)(2,3)(4,15)(5,16)(7,8)(9,13)(10,14)(11,12)(17,22)(18,20)(19,28)(21,29)(23,30)(24,25)(26,31)(27,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961675);