Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4419 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23);(2,6)(7,8)(25,26)(27,33)(28,32)(29,31)(30,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(1,4,5,3)(2,8,6,7)(11,18)(14,17)(19,24)(22,23)(26,31)(27,34)(32,38)(35,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(27,34)(30,33)(35,40)(37,39);(25,28,29,36)(26,32,31,38)(27,35,34,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615533);