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 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,12,19,16)(13,17,22,23)(18,20,21,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,3,5,8)(2,6,7,4)(10,15)(12,16)(17,23)(20,24);(1,3)(2,4)(5,8)(6,7)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,4)(2,3)(5,6)(7,8)(9,11,14,19)(10,16,15,12)(13,21,22,18)(17,20,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(9,13,14,22)(10,17,15,23)(11,18,19,21)(12,20,16,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358797);