Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
2088 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,13)(14,32)(15,19)(16,20)(18,27)(21,37)(22,26)(23,33)(24,34)(25,35)(28,30)(29,31)(36,38)(39,40);(2,5)(6,8)(9,11)(10,17)(12,26)(13,22)(14,24)(15,33)(16,25)(18,36)(19,23)(20,35)(21,28)(27,38)(29,40)(30,37)(31,39)(32,34);(9,12,16,29)(10,18,20,30)(11,21,25,38)(13,19,31,32)(14,28,15,27)(17,22,35,39)(23,36,34,37)(24,40,33,26);(1,3,4,7)(2,6,5,8)(9,13)(10,12)(11,22)(14,30)(15,18)(16,31)(17,26)(19,27)(20,29)(21,34)(23,38)(24,37)(25,39)(28,32)(33,36)(35,40);(9,10)(11,23)(12,18)(13,27)(14,19)(15,32)(16,20)(17,33)(21,36)(22,26)(24,35)(25,34)(28,31)(29,30)(37,38)(39,40)Polycyclic presentation:
2^9|c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482636);