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 | |
1952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,21)(14,19)(16,18)(17,28)(20,25)(22,31)(24,27)(26,36)(29,30)(32,34)(33,40)(35,38)(37,39);(2,5)(3,7)(9,11,15,10)(12,24,27,23)(13,21,16,18)(14,30,31,34)(17,25,20,28)(19,32,22,29)(26,33,39,35)(36,38,37,40);(9,12)(10,17)(11,20)(13,25)(14,26)(15,27)(16,28)(18,24)(19,33)(21,23)(22,35)(29,38)(30,36)(31,39)(32,40)(34,37);(2,5)(6,8)(9,13,15,16)(10,18,11,21)(12,25,27,28)(14,29,31,32)(17,24,20,23)(19,34,22,30)(26,38,39,40)(33,37,35,36);(1,3,4,7)(2,6,5,8)(9,14,15,31)(10,19,11,22)(12,26,27,39)(13,29,16,32)(17,33,20,35)(18,34,21,30)(23,36,24,37)(25,38,28,40)Polycyclic presentation:
2^9|b^2=i,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,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, 6580138);