Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | #53190 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,20)(19,23)(21,24)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(25,27,29,35)(26,30,31,33)(28,37,38,34)(32,40,36,39);(1,3,4,8)(2,5,6,7)(26,31)(30,33)(32,36)(39,40);(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,29,38)(26,32,31,36)(27,34,35,37)(30,39,33,40);(2,6)(5,7)(10,17)(11,20)(16,22)(19,21)(27,35)(30,33)(34,37)(39,40);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013095);