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 | #41994 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4306 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)(11,17)(12,22)(13,19)(14,20)(15,16)(18,21)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(10,16)(11,20)(12,21)(19,24)(28,36)(31,38)(35,39)(37,40);(9,11,15,20)(10,17,16,14)(12,13,21,23)(18,24,22,19)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,37,38,40);(1,3,5,4)(2,7,6,8)(9,12,15,21)(10,18,16,22)(11,13,20,23)(14,19,17,24)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,38)(34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,21)(12,20)(14,22)(15,23)(16,24)(17,18)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(9,14,15,17)(10,20,16,11)(12,19,21,24)(13,18,23,22)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,38)(36,40)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565949);