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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1948 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)(3,6)(4,8)(5,7)(10,13)(14,16)(17,18,21,23)(19,20,24,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,7)(5,8)(11,15)(14,16)(17,18,21,23)(19,20,24,22)(26,29)(31,33)(32,37)(38,39);(2,7)(3,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,23,20)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,22)(23,24)(25,29,30,26)(27,33,35,31)(28,32,36,37)(34,38,40,39)Polycyclic presentation:
2^9|a^2=fgi,b^2=fgi,c^2=fhi,e^2=f,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, 5691930);