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 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(3,9)(8,12)(10,15)(17,18,22,25)(19,20,23,27)(21,24,30,32)(26,28,29,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(1,2,6,4)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,19)(18,20)(21,29)(22,23)(24,31)(25,27)(26,30)(28,32)(34,37)(36,45)(39,41)(42,46);(3,9)(7,11)(10,15)(14,16)(19,23)(20,27)(26,29)(28,31)(33,35,38,43)(34,39,37,41)(36,42,45,46)(40,47,44,48);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,20,22,27)(18,23,25,19)(21,28,30,31)(24,29,32,26)(33,35)(34,39)(36,46)(37,41)(38,43)(40,48)(42,45)(44,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,48)(43,46);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,29)(25,32)(27,31)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,48,46,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748561);