Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,22)(12,17)(13,28)(14,20)(15,21)(16,24)(18,30)(19,26)(23,34)(25,32)(27,31)(29,36)(33,40)(35,38)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,24,15,12,26,25)(10,16,19,22,21,32,20,17)(13,27,30,40,29,23,38,39)(18,34,28,33,35,31,36,37);(2,7)(3,8)(13,29)(18,35)(23,27)(28,36)(30,38)(31,34)(33,37)(39,40);(1,4,5,6)(2,8,7,3)(9,12)(10,17)(11,15)(13,27)(14,25)(16,19)(18,33)(20,32)(21,22)(23,29)(24,26)(28,31)(30,40)(34,36)(35,37)(38,39);(9,13,26,38,15,29,14,30)(10,18,20,36,21,35,19,28)(11,23,25,40,12,27,24,39)(16,31,17,33,32,34,22,37)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7464575);