Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1452 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)(3,8)(4,6)(5,7)(9,10,14,16)(11,13,15,12)(25,26)(27,33)(28,36)(29,34)(30,31)(32,39)(35,40)(37,38);(1,3,5,6)(2,4,7,8)(9,11,14,15)(10,12,16,13)(18,21)(22,24)(26,31)(27,34)(32,36)(35,37);(3,6)(4,8)(9,12,14,13)(10,11,16,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,37)(30,34)(31,33)(32,38)(35,39)(36,40);(1,4,5,8)(2,6,7,3)(9,11)(10,12)(13,16)(14,15)(25,28)(26,32)(27,35)(29,40)(30,39)(31,36)(33,38)(34,37);(1,2)(3,4)(5,7)(6,8)(9,13,14,12)(10,15,16,11)(17,19,20,23)(18,22,21,24)(25,29)(26,27)(28,38)(30,33)(31,34)(32,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2655311);