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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(4,12)(7,10)(8,13)(18,23)(20,29)(24,27)(26,31)(33,34,39,45)(35,43,37,40)(36,42,47,41)(38,46,48,44);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,18,22,23)(19,27,28,24)(20,30,29,25)(21,26,32,31)(33,35,39,37)(34,40,45,43)(36,46,47,44)(38,41,48,42);(2,6)(3,9)(8,13)(11,14)(19,28)(21,32)(24,27)(26,31)(33,36,39,47)(34,41,45,42)(35,46,37,44)(38,40,48,43);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,19)(18,24)(20,31)(21,25)(22,28)(23,27)(26,29)(30,32)(33,36)(34,42)(35,44)(37,46)(38,43)(39,47)(40,48)(41,45);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,20,22,29)(18,25,23,30)(19,26,28,31)(21,24,32,27)(33,37,39,35)(34,43,45,40)(36,44,47,46)(38,42,48,41);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,21,22,32)(18,26,23,31)(19,25,28,30)(20,24,29,27)(33,38,39,48)(34,44,45,46)(35,41,37,42)(36,43,47,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840283);