Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,16)(11,14,12,15)(19,20)(23,24)(25,26)(27,32)(28,29)(30,33)(31,34)(35,39)(36,37)(38,40);(2,6)(3,7)(18,22)(23,24)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(3,7)(4,8)(9,11,13,12)(10,14,16,15)(17,18,21,22)(19,23,20,24)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,4)(2,3)(5,8)(6,7)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,24,22,23)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=gi,d^2=gh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5358364);