Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 | |
1830 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)(11,22)(12,29)(13,18)(14,19)(15,21)(16,26)(17,32)(20,31)(23,37)(24,35)(25,27)(28,38)(30,36)(33,34)(39,40);(2,6)(7,8)(9,11,14,26)(10,16,19,22)(12,30,13,24)(15,27,31,28)(17,37,34,39)(18,36,29,35)(20,25,21,38)(23,32,40,33);(1,3,5,4)(2,7,6,8)(9,12,31,33,14,13,15,32)(10,17,21,18,19,34,20,29)(11,23,28,24,26,40,27,30)(16,35,38,37,22,36,25,39);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,18,19,29)(11,24,26,30)(15,33,31,32)(16,36,22,35)(17,21,34,20)(23,28,40,27)(25,37,38,39);(9,10)(11,25)(12,17)(13,34)(14,19)(15,20)(16,28)(18,33)(21,31)(22,27)(23,39)(24,35)(26,38)(29,32)(30,36)(37,40)Polycyclic presentation:
2^9|b^2=i,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=dh,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, 6492004);