Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,7,5,8)(10,14)(12,15)(17,18,20,22)(19,21,23,24)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,40)(38,39);(1,3,6,5)(2,7,4,8)(10,14)(11,16)(19,23)(21,24)(26,27)(28,35)(29,37)(39,40);(1,3,6,5)(2,8,4,7)(9,10)(11,15)(12,16)(13,14)(17,19,20,23)(18,21,22,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,39,38,40);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,15,14,12)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,16,14,11)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,c^2=fgh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2997095);