Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(7,11)(10,12)(13,14)(15,20)(16,24)(17,19)(18,23)(21,26)(22,28)(25,27);(6,9)(7,11)(13,15,16,23,17,22,26,25)(14,18,21,20,19,27,24,28);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(21,24)(23,25);(6,9)(10,12)(13,15)(14,20)(16,25)(17,22)(18,21)(19,28)(23,26)(24,27);(5,7,8,11)(6,10,9,12)(14,19)(18,27)(20,28)(21,24)Polycyclic presentation:
2^9|b^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050868);