Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1924 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,18)(14,19)(15,21)(16,20)(17,25)(22,26)(23,27)(24,28)(29,34)(30,35)(31,39)(32,37)(33,38)(36,40);(2,5)(3,7)(9,11)(10,12)(13,17)(14,30)(15,23)(16,24)(18,25)(19,35)(20,28)(21,27)(22,32)(26,37)(29,39)(31,34)(33,36)(38,40);(9,12)(10,17)(11,21)(13,25)(14,31)(15,27)(16,28)(18,24)(19,36)(20,23)(22,34)(26,33)(29,40)(30,38)(32,35)(37,39);(2,5)(6,8)(9,13,16,15)(10,18,20,21)(11,17,24,23)(12,25,28,27)(14,29,33,32)(19,34,38,37)(22,30,39,36)(26,35,31,40);(1,3,4,7)(2,6,5,8)(9,14)(10,19)(11,22)(12,26)(13,29)(15,32)(16,33)(17,30)(18,34)(20,38)(21,37)(23,36)(24,39)(25,35)(27,40)(28,31)Polycyclic presentation:
2^9|d^2=i,e^2=f,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,e^c=ei,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580144);