Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,13,15)(11,12,16,14)(19,23)(22,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(2,3)(4,8)(9,11,13,16)(10,14,15,12)(26,30)(28,37)(31,33)(34,38);(2,3)(6,7)(10,15)(11,16)(17,18,21,20)(19,22,23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,37)(38,39);(1,3,5,2)(4,6,8,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,35)(26,31,30,33)(28,38,37,34)(32,40,36,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,15)(13,14)(17,20,21,18)(19,24,23,22)(25,28,29,37)(26,32,30,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=hi,d^2=gh,e^2=fghi,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, 3173211);