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 | #36745 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4183 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)(3,4)(5,7)(6,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,22)(20,24)(25,26)(27,34)(28,32)(29,37)(30,31)(33,39)(35,36)(38,40);(1,3)(2,6)(4,7)(5,8)(10,15)(12,20)(16,22)(18,21)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,39)(38,40);(10,15)(11,19)(17,24)(18,21)(26,31)(27,32)(33,37)(35,40);(9,11,14,19)(10,16,15,22)(12,18,20,21)(13,17,23,24)(25,27,30,32)(26,28,31,34)(29,35,39,40)(33,36,37,38);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,18)(13,22)(14,20)(15,24)(16,23)(19,21)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,23)(10,18,15,21)(11,17,19,24)(12,16,20,22)(25,29)(26,33)(27,35)(28,36)(30,39)(31,37)(32,40)(34,38)Polycyclic presentation:
2^9|d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9624742);