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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(11,12)(14,15)(17,18,21,22)(19,23,24,20)(25,26)(27,32)(28,36)(29,34)(30,31)(33,39)(35,38)(37,40);(9,10,13,16)(11,14,12,15)(19,24)(20,23)(26,31)(29,32)(33,36)(38,40);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,24)(22,23)(25,27,30,34)(26,32,31,29)(28,37,39,35)(33,38,36,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,23)(22,24)(25,28,30,39)(26,33,31,36)(27,35,34,37)(29,38,32,40);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,18)(19,20)(21,22)(23,24)(25,29,30,32)(26,34,31,27)(28,38,39,40)(33,37,36,35)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=ghi,d^2=f,e^2=ghi,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, 1901847);