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 | |
4455 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)(9,10,14,16)(11,13,18,15)(12,17,21,24)(19,22,20,23)(27,33)(31,34)(35,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,18)(10,15,16,13)(12,20,21,19)(17,22,24,23)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3,5,4)(2,7,6,8)(10,16)(11,18)(12,21)(22,23)(26,30)(31,34)(32,36)(38,40);(9,11)(10,15)(12,19)(13,16)(14,18)(17,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,24)(18,20)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,5,3)(2,8,6,7)(9,13,14,15)(10,11,16,18)(12,22,21,23)(17,19,24,20)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7897089);