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 | #34962 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4384 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,26)(27,33)(28,32)(29,31)(30,34)(35,40)(36,38)(37,39);(1,3,5,4)(2,7,8,6)(9,12,14,21)(10,16,17,20)(11,19,13,22)(15,23,18,24)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,37)(38,39);(1,3,5,4)(2,7,8,6)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(26,31)(27,34)(32,39)(35,36);(10,17)(15,18)(16,20)(23,24)(25,28,29,37)(26,32,31,39)(27,35,34,36)(30,38,33,40);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,e^2=i,f^2=gh,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, 7614880);