Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 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)(5,6,8,9)(7,10,11,12)(13,14,19,24)(15,20,26,28)(16,21,18,23)(17,22,25,27);(6,9)(10,12)(13,15,19,26)(14,20,24,28)(16,25,18,17)(21,27,23,22);(2,4)(13,16)(14,21)(17,25)(18,19)(22,27)(23,24);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,16)(18,26)(19,25)(20,21)(23,28)(24,27);(15,25)(16,18)(17,26)(20,27)(21,23)(22,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256930, #72256952, #72256992 |
| maximal normal subgroup of | #72256930, #72256952, #72256992 |
| maximal quotient of | #72264155, #72265136, #72265209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483358);