Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1808 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)(3,5)(4,6)(7,8)(9,10,15,19,14,21,13,20)(11,22,30,33,26,32,25,16)(12,18,27,17,29,36,28,34)(23,38,39,35,40,31,24,37);(1,2)(3,7)(4,6)(5,8)(9,11,15,30,14,26,13,25)(10,16,20,32,21,33,19,22)(12,23,28,24,29,40,27,39)(17,31,36,37,34,38,18,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,31)(18,19)(20,36)(21,34)(22,35)(24,30)(25,39)(26,40)(32,37)(33,38);(3,8)(5,7)(9,12)(10,18)(11,24)(13,28)(14,29)(15,27)(16,31)(17,19)(20,34)(21,36)(22,37)(23,30)(25,40)(26,39)(32,35)(33,38);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910253);