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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,36)(29,32)(30,33)(34,39)(35,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,31)(28,34)(29,37)(30,35)(32,38)(33,40)(36,39);(1,4,5,6)(2,7,8,3)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(25,28,30,29)(26,32,33,36)(27,34,35,37)(31,38,40,39);(3,7)(4,6)(10,15)(14,17)(16,22)(21,23)(25,29,30,28)(26,32,33,36)(27,34,35,37)(31,39,40,38);(2,8)(3,7)(11,18)(14,17)(19,24)(21,23)(26,33)(31,40)(32,36)(38,39);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,f^2=g,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, 8011472);