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 | |
4852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,19,22)(12,13,20,21)(17,18,23,24)(25,26,30,32)(27,31,35,37)(28,33,38,36)(29,39,40,34);(2,3)(6,7)(9,11,14,19)(10,15,16,22)(12,17,20,23)(13,18,21,24)(25,27,30,35)(26,31,32,37)(28,34,38,39)(29,36,40,33);(1,3,5,2)(4,7,8,6)(10,16)(12,20)(15,22)(17,23)(26,32)(27,35)(33,36)(34,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,13)(11,17)(14,20)(15,18)(16,21)(19,23)(22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,38)(39,40);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28)(26,33)(27,34)(29,31)(30,38)(32,36)(35,39)(37,40);(9,13)(10,12)(11,18)(14,21)(15,17)(16,20)(19,24)(22,23)(25,29)(26,34)(27,36)(28,37)(30,40)(31,38)(32,39)(33,35)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817198);