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 | |
1964 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)(3,5)(4,6)(7,8)(9,10)(11,20,24,30)(12,17)(13,18)(14,19)(15,22,16,23)(21,32,37,39)(25,31,36,29)(26,33)(27,34)(28,35)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,25)(13,23)(14,24)(16,19)(17,32)(18,30)(20,33)(21,27)(22,26)(28,36)(29,34)(31,40)(35,39)(37,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,24)(15,19)(17,31)(18,30)(20,33)(23,26)(25,27)(28,37)(29,35)(32,40)(34,39)(36,38);(3,8)(5,7)(9,12,26,38,14,28,13,27)(10,17,33,40,19,35,18,34)(11,21,23,36,24,37,22,25)(15,29,30,39,16,31,20,32);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,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, 6914397);