Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 |
| prime radical | #57564 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(3,7)(9,10)(11,19)(12,13)(14,17)(15,18)(16,23)(20,22)(21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,7)(2,6,4,8)(9,12,15,17)(10,13,18,14)(11,20,23,24)(16,21,19,22)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(1,2,5,4)(3,8,7,6)(9,13,15,14)(10,17,18,12)(11,21,23,22)(16,24,19,20)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39);(1,4,5,2)(3,8,7,6)(9,13,15,14)(10,12,18,17)(11,22,23,21)(16,24,19,20)(26,32)(30,33)(31,40)(38,39);(9,14,15,13)(10,17,18,12)(11,22,23,21)(16,24,19,20)Polycyclic presentation:
2^9|c^2=gi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012130);