Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4619 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,23)(20,22)(21,24)(25,26)(27,34)(28,37)(29,33)(30,32)(31,36)(35,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,20)(16,22,24,23)(25,27,30,36)(26,31,32,34)(28,38,29,35)(33,40,37,39);(2,6)(3,7)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,24)(18,23)(26,32)(31,34)(33,37)(39,40);(3,7)(5,8)(9,11,14,13)(10,15,17,18)(12,20,21,19)(16,23,24,22)(25,28,30,29)(26,33,32,37)(27,35,36,38)(31,39,34,40);(9,11,14,13)(10,15,17,18)(12,19,21,20)(16,22,24,23)(25,29,30,28)(26,33,32,37)(27,35,36,38)(31,40,34,39);(9,13,14,11)(10,15,17,18)(12,20,21,19)(16,22,24,23)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000278);