Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#537 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(5,6,8,10)(7,9,12,11)(15,21)(19,25)(22,27)(26,28);(5,6)(7,9)(8,10)(11,12)(14,18)(19,25)(20,23)(26,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28);(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533562);