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 | |
4129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(18,22)(19,25)(24,27)(26,31)(33,34)(35,40)(36,38)(37,46)(39,41)(42,45)(43,48)(44,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,35)(34,40)(36,45)(37,43)(38,42)(39,44)(41,47)(46,48);(2,6)(7,10)(8,12)(14,16)(19,25)(23,29)(26,31)(30,32)(34,41)(35,44)(36,43)(42,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36,39,43)(34,38,41,48)(35,45,44,37)(40,42,47,46);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,37,39,45)(34,42,41,46)(35,43,44,36)(38,47,48,40);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,38)(34,43)(35,46)(36,41)(37,47)(39,48)(40,45)(42,44)Polycyclic presentation:
2^9|b^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662840);