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 |
| prime radical | #40510 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4480 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)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26,30,32)(27,28,34,36)(29,31,39,38)(33,35,37,40);(1,2)(3,7)(4,8)(5,6)(10,15)(12,19)(17,21)(20,24)(25,27)(26,28)(29,37)(30,34)(31,40)(32,36)(33,39)(35,38);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,21)(18,23)(25,28,30,36)(26,27,32,34)(29,35,39,40)(31,33,38,37);(1,4,5,3)(2,8,6,7)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,38)(39,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,29)(26,31)(27,33)(28,35)(30,39)(32,38)(34,37)(36,40);(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,26)(27,28)(29,38)(30,32)(31,39)(33,40)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^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=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944195);