Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | |
2622 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,18)(14,19)(15,21)(16,20)(17,25)(22,26)(23,27)(24,28)(29,33)(30,34)(31,35)(32,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,17)(14,30)(15,23)(16,24)(18,25)(19,34)(20,28)(21,27)(22,31)(26,35)(29,38)(32,37)(33,39)(36,40);(9,12)(10,17)(11,21)(13,25)(14,26)(15,27)(16,28)(18,24)(19,30)(20,23)(22,35)(29,34)(31,40)(32,39)(33,38)(36,37);(2,6)(5,7)(9,13,16,15)(10,18,20,21)(11,17,24,23)(12,25,28,27)(14,29,32,31)(19,33,36,35)(22,30,38,37)(26,34,39,40);(3,8)(5,7)(9,14,16,32)(10,19,20,36)(11,22,24,38)(12,26,28,39)(13,29,15,31)(17,30,23,37)(18,33,21,35)(25,34,27,40)Polycyclic presentation:
2^9|d^2=i,e^2=i,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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574949);