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 | |
4154 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)(17,18)(19,21)(20,27)(22,23)(24,29)(25,30)(26,32)(28,31)(33,34)(35,44)(36,40)(37,42)(38,43)(39,45)(41,46)(47,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18)(19,21)(20,24)(22,23)(25,30)(26,28)(27,29)(31,32)(33,35,38,45)(34,39,43,44)(36,42,46,48)(37,40,47,41);(2,6)(7,10)(8,12)(14,16)(18,23)(19,25)(24,27)(26,31)(33,36,38,46)(34,40,43,41)(35,42,45,48)(37,39,47,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36)(34,41)(35,42)(37,39)(38,46)(40,43)(44,47)(45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,27)(25,31)(30,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,47)(43,44)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,45)(43,48)(44,46)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,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, 8659163);