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 | |
2651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(9,10,15,19)(11,20,24,34)(12,25,27,32)(13,18,14,28)(16,23,31,21)(17,29,33,26)(22,30,38,40)(35,36,39,37);(1,2,4,3)(5,6)(7,8)(11,21)(12,26)(13,14)(16,20)(17,25)(18,28)(23,24)(27,29)(31,34)(32,33)(35,39)(36,37);(9,11,15,24)(10,16,19,31)(12,22,27,38)(13,21,14,23)(17,30,33,40)(18,34,28,20)(25,35,32,39)(26,36,29,37);(6,8)(9,12,15,27)(10,17,19,33)(11,22,24,38)(13,26,14,29)(16,30,31,40)(18,32,28,25)(20,35,34,39)(21,36,23,37);(1,3,4,2)(9,13,15,14)(10,18,19,28)(11,23,24,21)(12,26,27,29)(16,20,31,34)(17,32,33,25)(22,37,38,36)(30,35,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,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, 7426446);