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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,22,21)(19,25,27,23)(20,24,29,30)(26,32,28,31)(33,34)(35,41)(36,37)(38,42)(39,45)(40,47)(43,44)(46,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19)(18,23)(20,28)(21,25)(22,27)(24,32)(26,29)(30,31)(33,35)(34,39)(36,43)(37,46)(38,45)(40,48)(41,42)(44,47);(3,12)(6,15)(8,16)(10,13)(17,18,22,21)(19,23,27,25)(20,24,29,30)(26,31,28,32)(33,34,38,42)(35,39,45,41)(36,37,40,47)(43,46,48,44);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20)(18,24)(19,26)(21,30)(22,29)(23,31)(25,32)(27,28)(33,36)(34,37)(35,43)(38,40)(39,46)(41,44)(42,47)(45,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,21,22,18)(19,23,27,25)(20,24,29,30)(26,32,28,31)(33,37)(34,40)(35,44)(36,42)(38,47)(39,43)(41,48)(45,46);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,19)(18,25)(20,26)(21,23)(22,27)(24,32)(28,29)(30,31)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8314368);