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 | |
4527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,32)(29,34)(30,31)(35,37)(36,38)(39,40);(1,3)(2,4)(5,7)(6,8)(11,17)(15,21)(18,23)(22,24)(25,26)(27,29)(28,36)(30,31)(32,38)(33,34)(35,37)(39,40);(10,14)(15,21)(16,19)(22,24)(26,31)(27,34)(32,36)(35,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,32)(27,35)(29,39)(30,38)(31,36)(33,37)(34,40);(1,4,5,6)(2,7,8,3)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,40)(35,36)(38,39)Polycyclic presentation:
2^9|e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9621753);