Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3280 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,19,25,27)(16,21,26,24)(18,22,28,23);(1,2,4,3)(6,9)(10,12)(13,15,17,25)(14,19,20,27)(16,22,26,23)(18,21,28,24);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,20)(15,21)(16,19)(18,28)(24,25)(26,27);(13,16)(14,21)(15,23)(17,26)(18,19)(20,24)(22,25)(27,28);(15,24)(16,19)(18,28)(21,25)(22,23)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486475);