Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
1950 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)(11,23,28,39)(12,29,25,38)(13,19)(14,21)(15,20)(16,22)(17,27,36,26)(18,32,35,31)(24,34)(30,37)(33,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,30)(13,31)(14,27)(15,26)(16,28)(18,37)(19,38)(20,39)(21,23)(22,36)(24,25)(29,40)(32,33)(34,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,26)(14,31)(15,32)(16,25)(17,34)(19,39)(20,29)(21,38)(22,35)(23,40)(27,33)(28,30)(36,37);(3,8)(5,7)(9,13,14,30,16,33,15,24)(10,19,21,37,22,40,20,34)(11,25,26,32,28,12,27,31)(17,35,39,29,36,18,23,38);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,31,25,32)(13,30,33,24)(17,23,36,39)(18,29,35,38)(19,34,40,37)Polycyclic presentation:
2^9|a^2=fg,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914390);