Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,7)(9,10,13,14)(11,15,16,12)(17,18,21,19)(20,23,24,22)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,12,14,15)(17,19,21,18)(20,23,24,22)(28,36)(31,38)(34,40)(37,39);(2,6)(4,8)(9,12,13,15)(10,11,14,16)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,5,7)(2,4,6,8)(17,20)(18,22)(19,23)(21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,12)(13,16)(14,15)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=fgh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5395626);