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×#3284 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,17,21,26)(16,20,23,27)(22,24,25,28);(1,2,4,3)(6,9)(10,12)(14,19)(15,17)(16,23)(21,26)(22,28)(24,25);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15,18,21)(14,17,19,26)(16,24,23,28)(20,25,27,22);(13,16,18,23)(14,20,19,27)(15,22,21,25)(17,24,26,28);(13,17)(14,21)(15,19)(16,25)(18,26)(20,28)(22,23)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=i,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, 6486479);