Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1372 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)(4,5)(6,7)(11,15)(14,16)(18,22)(20,24)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,3)(2,5)(4,6)(7,8)(18,22)(19,23)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(2,7)(3,4)(9,10,13,12)(11,14,15,16)(17,18)(19,20)(21,22)(23,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,4,6,3)(2,8,7,5)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,5,6,8)(2,3,7,4)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,24)(22,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=fghi,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4505032);