Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1766 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18,21,22)(19,20,24,23)(25,26,31,29)(27,32,28,35)(30,33,38,37)(34,39,36,40);(2,6)(3,8)(9,11)(10,14)(12,16)(13,15)(18,22)(20,23)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20,21,23)(18,19,22,24)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(1,4)(2,8)(3,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,30,31,38)(26,33,29,37)(27,34,28,36)(32,39,35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4801791);