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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4414 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)(11,18)(12,17)(13,22)(14,15)(16,20)(19,24)(21,23)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,6)(7,8)(10,15)(16,18)(17,22)(23,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,16,15,18)(12,21,13,19)(17,24,22,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,15,22)(11,19,20,21)(16,23,18,24)(27,34)(28,36)(30,38)(31,40);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,15,22)(11,19,20,21)(16,24,18,23)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=g,e^2=g,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, 7613455);