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 | #35301 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4540 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)(11,13)(12,16)(14,15)(17,22)(18,20)(19,24)(21,23)(25,26)(27,32)(28,38)(29,34)(30,33)(31,35)(36,39)(37,40);(1,2,5,6)(3,7,4,8)(10,15)(11,17)(16,23)(18,19)(25,27,31,38)(26,32,35,28)(29,36,30,37)(33,40,34,39);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28)(26,27)(29,39)(30,40)(31,32)(33,36)(34,37)(35,38);(25,29,31,30)(26,33,35,34)(27,36,38,37)(28,39,32,40);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30,31,29)(26,34,35,33)(27,37,38,36)(28,40,32,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549843);