Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574063 |
| 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 | |
1807 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,18)(13,28)(14,19)(15,20)(17,22)(21,31)(23,33)(24,34)(25,35)(26,37)(27,38)(29,32)(30,36)(39,40);(2,6)(7,8)(9,11)(10,16)(12,24)(13,22)(14,29)(15,23)(17,28)(18,34)(19,32)(20,33)(21,30)(25,27)(26,39)(31,36)(35,38)(37,40);(1,3,5,4)(2,7,6,8)(9,11,13,22,15,23,14,29)(10,17,19,16,20,32,28,33)(12,21,26,24,27,39,30,25)(18,34,36,31,38,35,37,40);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,28)(18,35)(19,32)(20,33)(21,26)(24,27)(30,39)(31,37)(34,38)(36,40);(9,12)(10,18)(11,21)(13,26)(14,30)(15,27)(16,31)(17,34)(19,36)(20,38)(22,24)(23,39)(25,29)(28,37)(32,35)(33,40)Polycyclic presentation:
2^9|c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6492015);