Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,22,23,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(2,3)(6,7)(9,10,13,12)(11,14,16,15)(19,23)(22,24)(26,31)(27,33)(29,37)(36,39);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,16,15)(17,19,21,23)(18,22,20,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,20,21,18)(19,24,23,22)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgi,c^2=ghi,d^2=fghi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1922546);