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 | #50647 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10,14,16)(11,15,19,24)(12,20,22,17)(13,23,21,18)(27,33)(28,36)(31,34)(32,37);(1,2,5,3)(4,6,8,7)(9,11,14,19)(10,15,16,24)(12,21,22,13)(17,23,20,18)(25,26)(27,34)(28,37)(29,30)(31,33)(32,36)(35,39)(38,40);(1,2,5,3)(4,7,8,6)(10,16)(11,19)(12,22)(18,23)(26,30)(27,33)(32,37)(35,38);(2,3)(6,7)(9,12,14,22)(10,17,16,20)(11,13,19,21)(15,18,24,23)(25,27)(26,31)(28,38)(29,33)(30,34)(32,40)(35,36)(37,39);(1,3,5,2)(4,7,8,6)(9,13)(10,18)(11,12)(14,21)(15,17)(16,23)(19,22)(20,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,39)(33,38)(34,40);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840384);