Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2275 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,15)(16,25,27,21)(17,22,20,24)(18,23,28,26);(6,9)(10,12)(13,15,19,14)(16,23,27,26)(17,22,20,24)(18,25,28,21);(1,2,3,4)(13,16,19,27)(14,21,15,25)(17,18,20,28)(22,26,24,23);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,24)(18,28)(19,20)(23,26);(13,18)(14,23)(15,26)(16,20)(17,27)(19,28)(21,22)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490567);