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 | |
4539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,15)(12,21)(13,17)(14,18)(16,23)(19,20)(22,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,22)(13,23)(14,19)(16,17)(18,20)(21,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(11,19)(13,22)(15,20)(17,24)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(9,12)(10,16)(11,13)(14,23)(15,24)(17,20)(18,21)(19,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,23)(22,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,12)(14,22)(15,21)(16,20)(18,24)(19,23)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7714042);