Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(11,15)(14,16)(19,20)(23,24)(25,26,31,35)(27,33,28,32)(29,30,40,34)(36,37,38,39);(2,6)(3,8)(10,13)(14,16)(18,22)(23,24)(25,27,31,28)(26,32,35,33)(29,38,40,36)(30,37,34,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,20,24)(25,28,31,27)(26,33,35,32)(29,38,40,36)(30,37,34,39);(1,3,5,8)(2,7,6,4)(17,19,21,20)(18,23,22,24)(25,29)(26,34)(27,36)(28,38)(30,35)(31,40)(32,37)(33,39);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,30,31,34)(26,29,35,40)(27,37,28,39)(32,38,33,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4613103);