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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,4)(5,6,9,16)(12,15,14,13)(17,18)(19,24)(20,25)(21,30)(22,23)(26,31)(27,28)(29,32)(33,34,39,40)(35,41,45,47)(36,44,46,42)(37,38,43,48);(1,3)(2,8)(4,10)(5,14)(6,13)(7,11)(9,12)(15,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,32)(30,31)(34,40)(36,46)(37,43)(41,47);(3,11)(4,8)(5,9)(6,16)(18,23)(19,28)(20,29)(21,31)(33,35,39,45)(34,41,40,47)(36,37,46,43)(38,42,48,44);(1,4)(2,3)(5,15)(6,14)(7,8)(9,13)(10,11)(12,16)(17,19)(18,24)(20,30)(21,25)(22,28)(23,27)(26,29)(31,32)(33,36,39,46)(34,42,40,44)(35,43,45,37)(38,47,48,41);(1,5,7,9)(2,6,10,16)(3,12,11,14)(4,13,8,15)(17,20,22,29)(18,25,23,32)(19,26,28,30)(21,24,31,27)(33,37,39,43)(34,38,40,48)(35,36,45,46)(41,44,47,42);(1,6)(2,9)(3,13)(4,14)(5,10)(7,16)(8,12)(11,15)(17,21,22,31)(18,26,23,30)(19,25,28,32)(20,24,29,27)(33,38)(34,43)(35,44)(36,47)(37,40)(39,48)(41,46)(42,45)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970494);