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 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26)(27,32)(28,36)(29,33)(30,34)(31,38)(35,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27)(26,29)(28,35)(30,33)(31,37)(32,34)(36,39)(38,40);(1,3,5,4)(2,6,7,8)(10,14)(15,17)(16,20)(23,24);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,28,30,38)(26,31,34,36)(27,35,33,40)(29,37,32,39);(1,4,5,3)(2,6,7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,37)(38,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29,30,32)(26,33,34,27)(28,37,38,39)(31,40,36,35)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9613657);