Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(25,26,30,31)(27,34,32,28)(29,37,40,33)(35,36,38,39);(1,3,6,5)(2,7,4,8)(17,19,21,24)(18,22,23,20)(26,31)(27,32)(33,37)(35,38);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,16,15)(17,20)(18,19)(21,22)(23,24)(25,27,30,32)(26,28,31,34)(29,38,40,35)(33,39,37,36);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,32)(27,31)(29,39)(30,34)(33,35)(36,40)(37,38);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,29)(26,33)(27,35)(28,36)(30,40)(31,37)(32,38)(34,39)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=ghi,d^2=fghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3762148);