Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,15)(13,16)(17,18,21,22)(19,23,24,20)(25,26,29,30)(27,31,33,34)(28,32,36,38)(35,39,37,40);(3,4)(6,7)(9,10,12,14)(11,13,15,16)(19,24)(20,23)(26,30)(27,33)(28,36)(39,40);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,4,5,3)(2,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,36)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3077185);