Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,15,10,14,9,13)(7,16,17,20,18,11,19,12);(1,2)(3,4)(5,7)(6,11)(8,19)(9,17)(10,18)(12,13)(14,16)(15,20);(2,4)(5,7)(6,12)(8,19)(9,17)(10,18)(11,15)(13,16)(14,20);(6,13,14,15)(11,12,16,20)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216230, #72318197, #72318490, … |
| maximal normal subgroup of | #72216230, #72318197, #72318490, … |
| maximal quotient of | #72215834 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303307);