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 | |
3974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(3,4)(10,13)(11,12)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30);(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442061);