Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,20,23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,23)(18,20,22,24)(26,29)(27,34)(32,38)(35,36);(2,3)(6,7)(10,13)(14,15)(18,22)(19,23)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28,30,37)(26,32,29,38)(27,35,34,36)(31,39,33,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)(25,29,30,26)(27,31,34,33)(28,32,37,38)(35,40,36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4102833);