Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1986 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)(11,20)(12,25)(13,18)(14,29)(15,19)(16,23)(17,28)(21,38)(22,30)(24,33)(26,32)(27,36)(31,35)(34,39)(37,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,30)(17,34)(18,20)(19,33)(21,28)(22,29)(25,37)(27,40)(31,39)(32,36)(35,38);(9,12,14,28,13,27,15,31)(10,17,19,25,18,35,29,36)(11,21,24,40,23,39,30,26)(16,32,33,34,20,37,22,38);(9,10)(11,22)(12,17)(13,18)(14,19)(15,29)(16,24)(20,30)(21,38)(23,33)(25,28)(26,37)(27,35)(31,36)(32,40)(34,39);(1,3,4,7)(2,6,5,8)(11,23)(16,20)(21,39)(22,33)(24,30)(26,40)(32,37)(34,38)Polycyclic presentation:
2^9|c^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352404);