Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #23860 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19);(1,2)(3,4)(6,9)(10,12)(14,18)(19,20);(2,4)(14,19)(15,17)(18,20);(2,4)(13,15,16,17)(14,20,18,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256973, #72256990, #72257029, … |
| maximal normal subgroup of | #72256973, #72256990, #72257029, … |
| maximal quotient of | #72264323, #72264655, #72264673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347239);