Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22)(26,27)(29,37)(31,33)(35,38);(3,8)(6,7)(11,16)(13,15)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,40,38,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,22)(19,23)(21,24)(25,28,30,36)(26,31,27,33)(29,38,37,35)(32,40,34,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(25,29)(26,32)(27,34)(28,35)(30,37)(31,39)(33,40)(36,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=g,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5588625);