Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1418 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)(3,8)(4,7)(5,6)(11,15)(13,16)(19,20)(23,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(2,6)(4,8)(9,10,12,14)(11,13,15,16)(18,22)(23,24)(29,37)(31,38)(34,40)(36,39);(1,2,5,6)(3,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,20,24)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3)(2,4)(5,7)(6,8)(17,19,21,20)(18,23,22,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4)(2,7)(3,6)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,19)(18,24,22,23)(25,29)(26,31)(27,34)(28,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=fgi,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5815584);