Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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:
(1,2,5,3)(4,6,8,7)(9,10,14,16)(11,12,13,15)(19,23)(22,24)(25,26)(27,33)(28,36)(29,35)(30,31)(32,38)(34,37)(39,40);(9,11,14,13)(10,12,16,15)(18,21)(22,24)(26,31)(29,33)(32,36)(37,39);(2,3)(6,7)(9,11,14,13)(10,15,16,12)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,15)(10,11,16,13)(25,28,30,38)(26,32,31,36)(27,34,35,40)(29,37,33,39);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,16,12)(17,19,20,23)(18,22,21,24)(25,29)(26,27)(28,37)(30,33)(31,35)(32,34)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4887731);