Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(9,10,15,20)(11,16,23,36)(12,25,22,39)(13,28,21,38)(14,31,33,19)(17,29,35,27)(18,26,34,30)(24,40,32,37);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,26)(13,29)(14,32)(15,23)(17,38)(18,39)(19,40)(20,36)(21,27)(22,30)(24,33)(25,34)(28,35)(31,37);(9,12,15,22)(10,17,20,35)(11,21,23,13)(14,29,33,27)(16,34,36,18)(19,39,31,25)(24,30,32,26)(28,40,38,37);(6,8)(9,13,15,21)(10,18,20,34)(11,22,23,12)(14,26,33,30)(16,35,36,17)(19,38,31,28)(24,27,32,29)(25,40,39,37);(1,3,4,2)(11,23)(12,22)(16,36)(17,35)(24,32)(25,39)(27,29)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426445);