Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30483 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,25)(13,19)(14,20)(15,22)(16,21)(18,26)(23,28)(24,33)(27,39)(29,35)(30,36)(31,37)(32,38)(34,40);(1,2)(3,6)(4,8)(5,7)(9,11,16,25)(10,17,21,12)(13,23,15,18)(14,30,32,34)(19,28,22,26)(20,36,38,40)(24,29,39,31)(27,37,33,35);(9,12,16,17)(10,18,21,23)(11,22,25,19)(13,26,15,28)(14,27,32,33)(20,34,38,30)(24,37,39,35)(29,40,31,36);(1,3,5,4)(2,6,7,8)(9,13,16,15)(10,19,21,22)(11,23,25,18)(12,26,17,28)(14,31,32,29)(20,37,38,35)(24,34,39,30)(27,36,33,40);(1,4,5,3)(2,6,7,8)(9,14,16,32)(10,20,21,38)(11,24,25,39)(12,27,17,33)(13,29,15,31)(18,34,23,30)(19,35,22,37)(26,40,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583727);