Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(11,12)(14,15)(25,26)(27,33)(28,34)(29,37)(30,31)(32,40)(35,36)(38,39);(9,10,13,16)(11,14,12,15)(18,22)(23,24)(26,31)(27,34)(32,37)(35,38);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,18,21,22)(19,23,20,24)(25,27)(26,28)(29,38)(30,34)(31,33)(32,39)(35,40)(36,37);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24)(25,28)(26,27)(29,39)(30,33)(31,34)(32,38)(35,37)(36,40);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,24,22,23)(25,29)(26,32)(27,35)(28,36)(30,40)(31,37)(33,39)(34,38)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fghi,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3630792);