Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,14,13)(11,12,16,15)(19,23)(21,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,40)(38,39);(1,3,6,4)(2,7,5,8)(9,11,14,16)(10,15,13,12)(26,27)(28,35)(29,37)(39,40);(9,12)(10,16)(11,13)(14,15)(17,18,20,22)(19,21,23,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,40,38,39);(1,4,6,3)(2,8,5,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,28,30,35)(26,31,27,33)(29,38,37,36)(32,40,34,39);(1,5,6,2)(3,7,4,8)(9,13,14,10)(11,15,16,12)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2771624);