Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,25)(17,22)(18,20)(19,24)(23,28)(26,27);(5,7,8,12)(6,9,10,11)(15,23)(16,18)(19,27)(20,25)(21,28)(24,26);(1,2,3,4)(13,15,17,24)(14,19,22,21)(16,23,18,26)(20,28,25,27);(6,10)(9,11)(13,16,17,18)(14,20,22,25)(15,23,24,26)(19,28,21,27);(7,12)(9,11)(13,15,17,24)(14,21,22,19)(16,23,18,26)(20,27,25,28)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590547);