Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,23,21,20)(26,27)(28,35)(32,34)(36,37);(2,4)(6,8)(10,12)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26,30,27)(28,33,35,31)(29,34,38,32)(36,39,37,40);(3,7)(6,8)(9,11,13,16)(10,14,12,15)(17,20,22,23)(18,19,24,21)(25,27,30,26)(28,31,35,33)(29,34,38,32)(36,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,24)(22,23)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39);(1,4,5,2)(3,6,7,8)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,23,24,20)(25,29,30,38)(26,32,27,34)(28,36,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=g,e^2=f,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, 3014969);