Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | |
1982 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)(10,16,19,27)(11,20,22,28)(12,23)(18,33,34,24)(21,36,38,35)(25,26)(30,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,24)(13,27)(14,16)(15,19)(18,25)(20,31)(21,37)(22,32)(23,33)(26,34)(28,29)(30,35)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,17,19,32)(12,21,23,38)(13,20,14,28)(16,29,27,31)(18,30,34,39)(24,37,33,40)(25,35,26,36);(3,8)(5,7)(9,12,14,26,15,23,13,25)(10,18,27,33,19,34,16,24)(11,21,28,36,22,38,20,35)(17,30,31,40,32,39,29,37);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914425);