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 | |
4470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,21)(14,16)(15,19)(17,22)(18,23)(20,24)(25,26,30,32)(27,28,34,36)(29,31,39,38)(33,35,37,40);(1,3,5,8)(2,6,4,7)(9,11,14,19)(10,15,16,12)(13,18,22,24)(17,23,21,20)(25,27)(26,28)(29,37)(30,34)(31,40)(32,36)(33,39)(35,38);(1,4,5,2)(3,7,8,6)(10,16)(12,15)(17,21)(20,23)(25,28,30,36)(26,27,32,34)(29,35,39,40)(31,33,38,37);(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,24)(22,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,38)(39,40);(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,23)(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)(15,23)(16,21)(19,24)(25,26)(27,28)(29,38)(30,32)(31,39)(33,40)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,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, 7944190);